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Bartleby Sitemap - Textbook Solutions

All Textbook Solutions for Calculus Volume 2

[T] Find the length of the dashed zig-zag path in the following figure.[T] Find the total length of the dashed path in the following figure.[T] The Sierpinski triangle is obtained from a triangle by deleting the middle fourth as indicated in the first step, by deleting the middle fourths of the remaining three congruent triangles in the second step, and in general deleting the middle fourths of the remaining triangles in each successive step. Assuming that the original triangle is shown in the figure, find the areas of the remaining parts of the original triangle after N steps and find the total length of all of the boundary triangles after N steps.[T] The Sierpinski gasket is obtained by dividing the unit square into nine equal sub—squares, removing the middle square, then doing the same at each stage to the remaining sub—squares. The figure shows the remaining set after four iterations. Compute the total area removed after N stages, and compute the length the total perimeter of the remaining set after N stages.For each of the following sequences, if the divergence test applies, either state that limnan a does not exist or find limnan . If the divergence test does not apply, state why. 138. an=nn+2 For each of the following sequences, if the divergence test applies, either state that limnan a does not exist or find limnan . If the divergence test does not apply, state why. 139. an=n5n23 For each of the following sequences, if the divergence test applies, either state that limnan a does not exist or find limnan . If the divergence test does not apply, state why. 140. an=n3n2+2n+1 For each of the following sequences, if the divergence test applies, either state that limnan a does not exist or find limnan . If the divergence test does not apply, state why. 141. an=(2n+1)(n1)(n+1)2 For each of the following sequences, if the divergence test applies, either state that limnan a does not exist or find limnan . If the divergence test does not apply, state why. 142. an=(2n+1)2n(3 n 2+1)n For each of the following sequences, if the divergence test applies, either state that limnan a does not exist or find limnan . If the divergence test does not apply, state why. 143. an=2n3n/2 For each of the following sequences, if the divergence test applies, either state that limnan a does not exist or find limnan . If the divergence test does not apply, state why. 144. an=2n+3n10n/2 For each of the following sequences, if the divergence test applies, either state that limnan a does not exist or find limnan . If the divergence test does not apply, state why. 145. an=e2/n For each of the following sequences, if the divergence test applies, either state that limnan a does not exist or find limnan . If the divergence test does not apply, state why. 146. an=e2/n For each of the following sequences, if the divergence test applies, either state that limnan a does not exist or find limnan . If the divergence test does not apply, state why. 147. an=tann For each of the following sequences, if the divergence test applies, either state that limnan a does not exist or find limnan . If the divergence test does not apply, state why. 148. an=1cos2(1/n)sin2(2/n)For each of the following sequences, if the divergence test applies, either state that limnan a does not exist or find limnan . If the divergence test does not apply, state why.149. an=(11n)2n For each of the following sequences, if the divergence test applies, either state that limnan a does not exist or find limnan . If the divergence test does not apply, state why. 150. an=lnnn For each of the following sequences, if the divergence test applies, either state that limnan a does not exist or find limnan . If the divergence test does not apply, state why. 151. an=(lnn)2n State whether the given p -series converges. 152. n=11n State whether the given p-series converges. 153. n=11nn State whether the given p-series converges. 154. n=11 n 23 State whether the given p-series converges. 155. n=11 n 43 State whether the given p-series converges. 156. n=1nn2e State whether the given p-series converges. 157. n=1nn2e Use the integral test to determine whether the following sums converge. 158. n=11n+5 Use the integral test to determine whether the following sums converge. 159. n=11n+53 Use the integral test to determine whether the following sums converge. 160. n=21nlnn Use the integral test to determine whether the following sums converge. 161. n=2n1+n2 Use the integral test to determine whether the following sums converge. 162. n=2en1+e2n Use the integral test to determine whether the following sums converge. 163. n=22n1+n4 Use the integral test to determine whether the following sums converge. 164. n=22nnln2n Express the following sums as p -series and determine whether each converges. 165. n=12lnn(Hint:2lnn=1/nln2)Express the following sums as p -series and determine whether each converges. 166. n=13lnn(Hint:3lnn=1/nln3)Express the following sums as p -series and determine whether each converges. 167. n=1n22lnn Express the following sums as p -series and determine whether each converges. 168. n=1n32lnn Use the estimate RNNf(t)dtto find a bound for the remainder RN=n=1ann=1Nan, where an=f(n). 169. n=11001n2 Use the estimate RNNf(t)dt to find a bound for the remainder RN=n=1ann=1Nan , where an=f(n). 170. n=11001n3 Use the estimate RNNf(t)dt to find a bound for the remainder RN=n=1ann=1Nan , where an=f(n). 171. n=1100011+n2 Use the estimate RNNf(t)dt to find a bound for the remainder RN=n=1ann=1Nan , where an=f(n). 172. n=11000n/2n [T] Find the minimum value of N such that the remainder estimate N+1fRNNfguarantees that n=1Nan estimates n=1an. accurate to within the given error. 173. an=1n2,error10-4 [T] Find the minimum value of N such that the remainder estimate N+1fRNNfguarantees that n=1Nan estimates n=1an. accurate to within the given error. 174. an=1n1,1,error10-4 [T] Find the minimum value of N such that the remainder estimate N+1fRNNfguarantees that n=1Nan estimates n=1an. accurate to within the given error. 175. an=1nln1n,error10-3 [T] Find the minimum value of N such that the remainder estimate N+1fRNNfguarantees that n=1Nan estimates n=1an. accurate to within the given error. 176. an=1nln2n,error10-3 [T] Find the minimum value of N such that the remainder estimate N+1fRNNfguarantees that n=1Nan estimates n=1an. accurate to within the given error. 177. an=11+n2,error10-3 In the following exercises, find a value of N such that RN is smaller than the desired error. Compute the corresponding sum n=1Nan and compare it to the given estimate of the infinite series. 178. an=1n11error10-4n=11n11=1.000494...In the following exercises, find a value of N such that RN is smaller than the desired error. Compute the corresponding sum n=1Nan and compare it to the given estimate of the infinite series. 179. an=1en,error10-5,n=11en=1e1=0.581976...In the following exercises, find a value of N such that RN is smaller than the desired error. Compute the corresponding sum n=1Nan and compare it to the given estimate of the infinite series. 180. an=1en2,error10-5,n=1n/en2=0.40488139857...In the following exercises, find a value of N such that RN is smaller than the desired error. Compute the corresponding sum n=1Nan and compare it to the given estimate of the infinite series. 181. an=1/n4,error10-5,n=1n/en2=4/90=1.08232...In the following exercises, find a value of N such that RN is smaller than the desired error. Compute the corresponding sum n=1Nan and compare it to the given estimate of the infinite series. 182. an=1/n6,error10-6,n=11/n4=6/945=1.01734306...Find the limit as n of 1n+1n+1+...+12n . (Hint: Compare to n2n1tdt.)184. Find the limit as n of 1n+1n+1+...+13n The next few exercises are intended to give a sense of applications in which partial sums of the harmonic series arise. 185. In certain applications of probability, such as the so—called Watterson estimator for predicting mutation rates in population genetics, it is important to have an accurate estimate of the number Hk=(1+12+13+...+1k) Recall that Tk=Hklnk is decreasing. Compute T=limkTkto four decimal places. (Hint: 1k+1kk+11xdx .)The next few exercises are intended to give a sense of applications in which partial sums of the harmonic series arise. 186. [T] Complete sampling with replacement, sometimes called the coupon collector ‘s problem, is phrased as follows: Suppose you have N unique items in a bin. At each step, an item is chosen at random, identified, and put back in the bin. The problem asks what is the expected number of steps E(N) that it takes to draw each unique item at least once. It turns out that E(N)=N,HN=N(1+12+13+...+1N) . Find E(N) for N = 10. 20. and 50.The next few exercises are intended to give a sense of applications in which partial sums of the harmonic series arise. 188. Suppose a scooter can travel 100 km on a full tank of fuel. Assuming that fuel can be transferred from one scooter to another but can only be carried in the tank, present a procedure that will enable one of the scooters to travel I 100HNkm, where HN= I + 1/2 + …+ 1/N.The next few exercises are intended to give a sense of applications in which partial sums of the harmonic series arise. 189. Show that for the remainder estimate to apply on [N, ) it is sufficient that f(x) be decreasing on [N. ), but f need not be decreasing on 11, ).The next few exercises are intended to give a sense of applications in which partial sums of the harmonic series arise. 190. [T] Use the remainder estimate and integration by parts to approximate n=1n/en within an error smaller than 0.000 1.The next few exercises are intended to give a sense of applications in which partial sums of the harmonic series arise. 191. Does , n=11n( lnn)p converge if p is large enough? If so, for which p?The next few exercises are intended to give a sense of applications in which partial sums of the harmonic series arise. 192. [T] Suppose a computer can sum one million terms per second of the divergent series n=1N1n -. Use the integral test to approximate how many seconds it will take to add up enough terms for the partial sum to exceed 100.The next few exercises are intended to give a sense of applications in which partial sums of the harmonic series arise. 193. [T] A fast computer can sum one million terms pet second of the divergent series Use the integral n=1N12lnntest to approximate how many seconds it will take to add up enough terms for the partial sum to exceed 100.Use the comparison test to determine whether the following series converge. 194. n=1anwherean=2n(n+1)Use the comparison test to determine whether the following series converge. 195. n=1anwherean=2n(n+1/2)Use the comparison test to determine whether the following series converge. 196. n=112(n+1)Use the comparison test to determine whether the following series converge. 197. n=11( nlnn)2 Use the comparison test to determine whether the following series converge. 198. n=11( nlnn)2 Use the comparison test to determine whether the following series converge. 199. n=1n!(n+2)!Use the comparison test to determine whether the following series converge. 200. n=11n!Use the comparison test to determine whether the following series converge. 201. n=1sin(1/n)n Use the comparison test to determine whether the following series converge. 202. n=1sin2nn2 Use the comparison test to determine whether the following series converge. 203. n=1sin2(1/n)n Use the comparison test to determine whether the following series converge. 204. n=1n1,21n2,3+1 Use the comparison test to determine whether the following series converge. 205. n=1n+1nn Use the comparison test to determine whether the following series converge. 206. n=1n4 n 4+ n 23 Use the limit comparison test to determine whether each of the following series converges or diverges. 207. n=1( lnn n)2 Use the limit comparison test to determine whether each of the following series converges or diverges. 208. n=1( lnn n 0.6)2 Use the limit comparison test to determine whether each of the following series converges or diverges. 209. n=1ln(1+ 1 n)n Use the limit comparison test to determine whether each of the following series converges or diverges. 210. n=1ln(1+1 n 2)Use the limit comparison test to determine whether each of the following series converges or diverges. 211. n=114n3n Use the limit comparison test to determine whether each of the following series converges or diverges. 212. n=11n2nsinn Use the limit comparison test to determine whether each of the following series converges or diverges. 213. n=11e( 1.1)n3n Use the limit comparison test to determine whether each of the following series converges or diverges. 214. n=11e( 1.01)n3n Use the limit comparison test to determine whether each of the following series converges or diverges. 215. n=11n1+1/n Use the limit comparison test to determine whether each of the following series converges or diverges. 216. n=1121+1/nn1+1/n Use the limit comparison test to determine whether each of the following series converges or diverges. 217. n=1(1nsin( 1 n))Use the limit comparison test to determine whether each of the following series converges or diverges. 218. n=1(1cos( 1 n))Use the limit comparison test to determine whether each of the following series converges or diverges. 219. n=11n(tan1n2)Use the limit comparison test to determine whether each of the following series converges or diverges. 220. n=1(1 1 n)n,n (Hint: (11n)n1/e )Use the limit comparison test to determine whether each of the following series converges or diverges. 221. n1(1e1/n) (Hint: 1/e(11/n)n , so If so, for which p?Use the limit comparison test to determine whether each of the following series converges or diverges. 222. Does n=11( lnn)p converge if p is large enough? If so, for which p?Use the limit comparison test to determine whether each of the following series converges or diverges. 223. Does n=1( lnn n)p converge if p is large enough? If so, for which p?Use the limit comparison test to determine whether each of the following series converges or diverges. 224. For which p does the series n=12pn/3n Use the limit comparison test to determine whether each of the following series converges or diverges. 225. 225. For which p> 0 does the series n=1np2n converge?Use the limit comparison test to determine whether each of the following series converges or diverges. 226. For which r> 0 does the series n=1r n p2n converge?Use the limit comparison test to determine whether each of the following series converges or diverges. 227. For which r > 0 does the series n=12nr n 2 converge?Use the limit comparison test to determine whether each of the following series converges or diverges. 228. Find all values of p and q such that n=1np( n!)q converges.Use the limit comparison test to determine whether each of the following series converges or diverges. 229. Does n=1sin2(nr/2)n converge or diverge? Explain.Use the limit comparison test to determine whether each of the following series converges or diverges. 230. Explain why, for each n, at lest one of {|sin n|, |sin(n+1),...,|sin n+6|} is larger than ½. Use this relation to test convergence of n=1sinnn Use the limit comparison test to determine whether each of the following series converges or diverges. 231. Suppose that an0and bn0 and that n=1a2n and n=1b2n converge. Prove that n=1anbnconverges and n=1anbn12( n=1 a n 2+ n=1 b n 2) .Use the limit comparison test to determine whether each of the following series converges or diverges. 232. Does n=12lnlnn converge? (Hint: Write 2lnln n as a power of ln n.)Use the limit comparison test to determine whether each of the following series converges or diverges. 233. Does n=12lnlnn converge? (Hint: Use t=eln(t) to compare to a p-series.)Use the limit comparison test to determine whether each of the following series converges or diverges. 234. Does n=2(lnn)lnlnn converge? (Hint: Compare an to 1/n ).Use the limit comparison test to determine whether each of the following series converges or diverges. 235. Show that if an0 and n=1an converges, then n=1a2n converges. If n=1a2n converges, does n=1an necessarily converge?Use the limit comparison test to determine whether each of the following series converges or diverges. 236. Suppose that an0 for all n=1an and that converges. Suppose that b is an arbitrary sequence of zeros and ones. Does n=1anbnnecessarily converge?Use the limit comparison test to determine whether each of the following series converges or diverges. 237. Suppose that an0 for all ii and that n=1an diverges. Suppose that bnis an arbitrary sequence of zeros and ones with infinitely many terms equal to one. Does n=1anbnnecessarily diverge?Use the limit comparison test to determine whether each of the following series converges or diverges. 238. Complete the details of the following argument: If n=11n converges to a finite sum s. then 12s=12+14+16+... and s12s=1+13+15+... Why does this lead to a contradiction?Use the limit comparison test to determine whether each of the following series converges or diverges. 239. Show that if an0 and n=1a2n converges, then n=1sin2(an) converges.Use the limit comparison test to determine whether each of the following series converges or diverges. 240. Suppose that an/bn0 in the comparison test, where an0 and bn0 . Prove that if bn converges, then an converges.Use the limit comparison test to determine whether each of the following series converges or diverges. 241. Let b be an infinite sequence of zeros and ones.What is the largest possible value of x=n=1bn/2n?Use the limit comparison test to determine whether each of the following series converges or diverges. 242. Let bn be an infinite sequence of zeros and ones. What is the largest possible value of x=n=1dn/10n that converges?wUse the limit comparison test to determine whether each of the following series converges or diverges. 243. Explain why, if x >1/2. then x cannot be written z=n=2nbn2(bn=0or1,b1=0)..[T] Evelyn has a perfect balancing scale, an unlimited number of 1 -kg weights, and one each of 1/2 -kg. 1/4 -kg. 1/8 -kg, and so on weights. She wishes to weigh a meteorite of unspecified origin to arbitrary precision. Assuming the scale is big enough, can she do it? What does this have to do with infinite series?[T] Robert wants to know his body mass to arbitraiy precision. He has a big balancing scale that works perfectly, an unlimited collection of 1 —kg weights, and nine each of 0.1 -kg. 0.01 -kg. 0.001 -kg. and so on weights. Assuming the scale is big enough, can lie do this? What does this have to do with infinite series?The series n=112n is half the harmonic series and hence diverges. It is obtained from the harmonic series by deleting all terms in which ii is odd. Let in > 1 be fixed. Slios’, more generally, that deleting all terms 1/n where n = mk for some integer k also results in a divergent series.In view of the previous exercise, it may be surprising that a subseries of the harmonic series in which about one in every five terms is deleted might converge. A depleted harmonic series is a series obtained from n=11n - by removing any term 1/n if a given digit, say 9. appears in the decimal expansion of is. Argue that this depleted harmonic series converges by answering the following questions. a. How many whole numbers ii have d digits? b. How many d-digit whole numbers h(d). do not contain 9 as one or more of their digits? c. That is the smallest d-digit number m(d)? d. Explain why the deleted harmonic series is bounded by d=1h(d)m(d) . e. Show that d=1h(d)m(d) converges.Suppose that a sequence of numbers an> 0 has the property that a1= 1 and an+1=1n+1Sn . where Sn=a1+...+an . Can you determine whether n=1an converges? (Him: Snis monotone.)Suppose that a sequence of numbers a > 0 has the property that a1=1 and an+1=1(n+1)2Sn. where Sn=a1+...+an. Can you determine whether n=1an converges? (Him: S2=a2+a1=a2+S1=a2+1=1+1/4=(1+1/4)S1,S3=132S2+S2=(1+1/9)S2=(1+1/9)(1+1/4)S1, etc. Look at ln(Sn) , and use ln(1+t)t,t0.)State whether each of the following series converges absolutely, conditionally, or not at all. 250. n=1(1)n+1nn+3 State whether each of the following series converges absolutely, conditionally, or not at all. 251. n=1(1)n+1n+1n+3 State whether each of the following series converges absolutely, conditionally, or not at all. 252. n=1(1)n+11n+3 State whether each of the following series converges absolutely, conditionally, or not at all. 253. n=1(1)n+1n+3n State whether each of the following series converges absolutely, conditionally, or not at all. 254. n=1(1)n+11n!State whether each of the following series converges absolutely, conditionally, or not at all. 255. n=1(1)n+13nn!State whether each of the following series converges absolutely, conditionally, or not at all. 256. n=1(1)n+1( n1 n)n State whether each of the following series converges absolutely, conditionally, or not at all. 257. n=1(1)n+1( n+1 n)n State whether each of the following series converges absolutely, conditionally, or not at all. 258. n=1(1)n+1sin2n State whether each of the following series converges absolutely, conditionally, or not at all. 259. n=1(1)n+1cos2n State whether each of the following series converges absolutely, conditionally, or not at all. 260. n=1(1)n+1sin2(1/n)State whether each of the following series converges absolutely, conditionally, or not at all. 261. n=1(1)n+1cos2(1/n)State whether each of the following series converges absolutely, conditionally, or not at all. 262. n=1(1)n+1ln(1+1n)State whether each of the following series converges absolutely, conditionally, or not at all. 263. n=1(1)n+1ln(1+1n)State whether each of the following series converges absolutely, conditionally, or not at all. 264. n=1(1)n+1n21+n4 State whether each of the following series converges absolutely, conditionally, or not at all. 265. n=1(1)n+1ne1+n State whether each of the following series converges absolutely, conditionally, or not at all. 266. n=1(1)n+12l/n State whether each of the following series converges absolutely, conditionally, or not at all. 267. n=1(1)n+1n1/n State whether each of the following series converges absolutely, conditionally, or not at all. 268. n=1(1)n(1n1/n) (Hint: cos (1/n)11/n2 for large n.)State whether each of the following series converges absolutely, conditionally, or not at all. 269. n=1(1)n+1n(1cos( 1 n)) (Hint: cos (1/n)11/n2 for large n.)State whether each of the following series converges absolutely, conditionally, or not at all. 270. n=1(1)n+1(n+1n) (Hint: Rationalize the numerator.State whether each of the following series converges absolutely, conditionally, or not at all. 271. n=1(1)n+1(1 n1 n+1) (Hint: Crossmultiply then rationalize numerator.)State whether each of the following series converges absolutely, conditionally, or not at all. 272. n=(1)n+1(tan1( tan 1n)) (Hint: Use Mean Value Theorem.)State whether each of the following series converges absolutely, conditionally, or not at all. 273. n=1(1)n+1n(tan1(n+1)tan1n) (Hint: Use Mean Value Theorem.)State whether each of the following series converges absolutely, conditionally, or not at all. 274. n=1(1)n+1(1n1n+1)State whether each of the following series converges absolutely, conditionally, or not at all. 275. n=1(1)n+1(1n1n+1)State whether each of the following series converges absolutely, conditionally, or not at all. 276. n=1cos(n)n State whether each of the following series converges absolutely, conditionally, or not at all. 277. n=1cos(n)n1/n State whether each of the following series converges absolutely, conditionally, or not at all. 278. n=11nsin(n2)State whether each of the following series converges absolutely, conditionally, or not at all. 279. n=1sin(n/2)sin(1/n)In each of the following problems, use the estimate RNbN+1to find a value of N that guarantees that the sum of the first N terms of the alternating series n=1(1)n+1bndiffers from the infinite sum b at most the given error. Calculate the partial sum SN for this N. 280. [T]bn=1/n,error10-5 In each of the following problems, use the estimate RNbN+1to find a value of N that guarantees that the sum of the first N terms of the alternating series n=1(1)n+1bndiffers from the infinite sum by at most the given error. Calculate the partial sum SN for this N. 281. [T]bn=1/ln(n),n2,error10-1 In each of the following problems, use the estimate RNbN+1to find a value of N that guarantees that the sum of the first N terms of the alternating series n=1(1)n+1bndiffers from the infinite sum by at most the given error. Calculate the partial sum SN for this N. 282. [T]bn=1/n,error10-3 In each of the following problems, use the estimate RNbN+1to find a value of N that guarantees that the sum of the first N terms of the alternating series n=1(1)n+1bndiffers from the infinite sum by at most the given error. Calculate the partial sum SN for this N. 283. [T]bn=1/2n,error10-6 In each of the following problems, use the estimate RNbN+1to find a value of N that guarantees that the sum of the first N terms of the alternating series n=1(1)n+1bndiffers from the infinite sum by at most the given error. Calculate the partial sum SN for this N. 284. [T]bn=ln(1+1n),error10-3 In each of the following problems, use the estimate RNbN+1to find a value of N that guarantees that the sum of the first N terms of the alternating series n=1(1)n+1bndiffers from the infinite sum by at most the given error. Calculate the partial sum SN for this N. 285. [T]bn=1/n2,error10-6 For the following exercises, indicate whether each of the following statements is true or false. If the statement is false, provide an example in which it is false. 286. If bn0 is decreasing and limnbn=0 then limnbn=0converges absolutely.For the following exercises, indicate whether each of the following statements is true or false. If the statement is false, provide an example in which it is false. 287. If bn0 is decreasing then limnbn=0 , then n=1(b2n1bn) converges absolutely.For the following exercises, indicate whether each of the following statements is true or false. If the statement is false, provide an example in which it is false. 288. If bn0 and limnbn=0 then n=1(12(b3n2+b3n1)b3n converges.For the following exercises, indicate whether each of the following statements is true or false. If the statement is false, provide an example in which it is false. 289. If bn0 is decreasing and n1(b3n2+b3n1b3n) converges then n=1b3n2 converges.For the following exercises, indicate whether each of the following statements is true or false. If the statement is false, provide an example in which it is false. 290. If bn0 is decreasing and n=1(b3n2+b3n1b3n) converges conditionally but not absolutely, then bn does not tend to zero.For the following exercises, indicate whether each of the following statements is true or false. If the statement is false, provide an example in which it is false. 291. Let an+=an if an0 and an=an if converges conditionally but not absolutely, then bn does not tend to zero.For the following exercises, indicate whether each of the following statements is true or false. If the statement is false, provide an example in which it is false. 292. Suppose that an is a sequence of positive real numbers and that n1an+ converge. Suppose that bn is an arbitrary sequence of ones minus ones . Does n=1anbn necessarily converge?The following series do not satisfy the hypotheses of the alternating series test as stated. In each case, state which hypothesis is not satisfied. State whether the series converges absolutely. 294. n=1(1)n+1 sin 2nn The following series do not satisfy the hypotheses of the alternating series test as stated. In each case, state which hypothesis is not satisfied. State whether the series converges absolutely. 294. n=1(1)n+1 sin 2nn The following series do not satisfy the hypotheses of the alternating series test as stated. In each case, state which hypothesis is not satisfied. State whether the series converges absolutely. 295. n=1(1)n+1 cos 2nn The following series do not satisfy the hypotheses of the alternating series test as stated. In each case, state which hypothesis is not satisfied. State whether the series converges absolutely. 296. 1+1213+14+15+161718+...The following series do not satisfy the hypotheses of the alternating series test as stated. In each case, state which hypothesis is not satisfied. State whether the series converges absolutely. 297. 1+1213+14+1516+17+1819+...The following series do not satisfy the hypotheses of the alternating series test as stated. In each case, state which hypothesis is not satisfied. State whether the series converges absolutely. 298. Show that the alternating series 112+1214+1316+1418+... does not converge. What hypothesis of the alternating series test is not met?The following series do not satisfy the hypotheses of the alternating series test as stated. In each case, state which hypothesis is not satisfied. State whether the series converges absolutely. 299. Suppose that an converges absolutely. Show that the seres consisting of the positive terms an also converges.The following series do not satisfy the hypotheses of the alternating series test as stated. In each case, state which hypothesis is not satisfied. State whether the series converges absolutely. 300. Show that the alternating series 2335+4759+... does not converge. What hypothesis of the alternating series test is not met?The following series do not satisfy the hypotheses of the alternating series test as stated. In each case, state which hypothesis is not satisfied. State whether the series converges absolutely. 301. Show that the alternating series cos=122!+44!+66!+... will be derived in the next chapter. Use the remainder RNbN+1 to find a bound for the error in estimating cos by the fifth partial sum 12/2!+4/4!6/6!+8/8! for =1. What =/6 , = .The following series do not satisfy the hypotheses of the alternating series test as stated. In each case, state which hypothesis is not satisfied. State whether the series converges absolutely. 302. The formula sin=33!+55!+77!+... will be derived in the next chapter. Use the remainder RNbN+1 to find a bound for the error in estimating sin by the fifth partial sum 3/3!+5/5!7/7!+9/9! for =1. =/6 , and = .The following series do not satisfy the hypotheses of the alternating series test as stated. In each case, state which hypothesis is not satisfied. State whether the series converges absolutely. 303. How many terms in cos=122!+44!+66!+... are needed to approximate cos 1 accurate to an error of at most 0.00001?The following series do not satisfy the hypotheses of the alternating series test as stated. In each case, state which hypothesis is not satisfied. State whether the series converges absolutely. 304. How many terms in sin =33!+55!+77!+... are needed to approximate sin 1 accurate sin 1 accurate to an error of at most 0.00001?The following series do not satisfy the hypotheses of the alternating series test as stated. In each case, state which hypothesis is not satisfied. State whether the series converges absolutely. 305. Sometimes the alternating series n=(1)n1bn converges to a certain fraction of an absolutely convergent series n=1bnat a faster rate. Given that n=11n226 , find S=1122+132+142+... Which of the series 6n=11n2and Sn=1( 1)n1n2 gives a better estimation of 2 using 1000 terms?The following alternating series converge to given multiples of . Find the value of N predicted by the remainder estimate such that the Nth partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum N for which the error bound holds, and give the desired approximate value in each case. Up to 15 decimals places, =3.141592653589793… 306. [T]4=n=0( 1)n2n+1,error0.0001 The following alternating series converge to given multiples of . Find the value of N predicted by the remainder estimate such that the Nth partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum N for which the error bound holds, and give the desired approximate value in each case. Up to 15 decimals places, =3.141592653589793… 307. [T]12n=0( 1)n2n+1,error0.0001 The following alternating series converge to given multiples of . Find the value of N predicted by the remainder estimate such that the Nth partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum N for which the error bound holds, and give the desired approximate value in each case. Up to 15 decimals places, =3.141592653589793… 308. [T] The series n=0sin(x+n)x+nplays an important role in signal processing. Show that E n=0sin(x+n)x+nconverges whenever 0 . (Hint: Use the formula for the sum of a sum of angles.)The following alternating series converge to given multiples of . Find the value of N predicted by the remainder estimate such that the Nth partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum N for which the error bound holds, and give the desired approximate value in each case. Up to 15 decimals places, =3.141592653589793… 309. [T]ifn=1N(1)n11nln2,what is 1+13+15121416+17+19+11118110112+...?The following alternating series converge to given multiples of . Find the value of N predicted by the remainder estimate such that the Nth partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum N for which the error bound holds, and give the desired approximate value in each case. Up to 15 decimals places, =3.141592653589793... 310. [T] Plot the series n=1100cos(2nx)n for 0x1 . Explain why n=1100cos(2nx)n diverges when x = 0. 1. How does the series behave for other x?The following alternating series converge to given multiples of . Find the value of N predicted by the remainder estimate such that the Nth partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum N for which the error bound holds, and give the desired approximate value in each case. Up to 15 decimals places, =3.141592653589793... 31. [T]Plottheseriesn=1100sin(2nx)n for 0 x<1 The following alternating series converge to given multiples of . Find the value of N predicted by the remainder estimate such that the Nth partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum N for which the error bound holds, and give the desired approximate value in each case. Up to 15 decimals places, =3.141592653589793... 312. [T] Plot the series n=1100cos(2nx)n for 0x1 and describe its graph.The following alternating series converge to given multiples of . Find the value of N predicted by the remainder estimate such that the Nth partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum N for which the error bound holds, and give the desired approximate value in each case. Up to 15 decimals places, =3.141592653589793...The following alternating series converge to given multiples of . Find the value of N predicted by the remainder estimate such that the Nth partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum N for which the error bound holds, and give the desired approximate value in each case. Up to 15 decimals places, =3.141592653589793... 315. [T] The Euler transform rewrites = n=O )“b,, as S= (—1)’2” I (Z1)b_1. For the n=O alternating harmonic series, it takes the form -I —‘ (—1)” 1 In(2) = = L ,.• Compute partial n=I n=I sums of ,, until the’ approximate in(2) accurate n=I fl2 to within 0.0001. How many terms are needed? Compare this answer to the number of terms of the alternating harmonic series are needed to estimate ln(2).Series Converging to and 1/ Dozens of series exist that converge to or an algebraic expression containing . Here we look at several examples and compare their rates of convergence. By rate of conveigence. we mean the number of terms necessazy for a partial sum to be within a certain amount of the actual value. The series representations of z in the first two examples can be explained using Maclaunn series, which axe discussed in the next chapter. The third example relies on material beyond the scope of this text. 1. The series =4n=1( 1)n+12n1=443+4547+49... was discovered by Gregory and Leibniz in the late 1600s. This result follows from the Maclaurin series for f(x) = tan-1 x. We will discuss this series in the next chapter. a. Prove that this series converges. b. Evaluate the partial sums Snfor n = 10. 20. 50. 100. C. Use the iemainder estimate for alternating series to get a bound on the error Rn. d. What is the smallest value of N that guarantees |RN| <0.01? Evaluate SN.Series Converging to and 1/ Dozens of series exist that converge to or an algebraic expression containing . Here we look at several examples and compare their rates of convergence. By rate of conveigence. we mean the number of terms necessazy for a partial sum to be within a certain amount of the actual value. The series representations of z in the first two examples can be explained using Maclaunn series, which axe discussed in the next chapter. The third example relies on material beyond the scope of this text. 2. The series =6n=0( 2n)! 2 4n+1( n!)( 2n+1)=6( 1 2+ 1 2.3 ( 1 2 ) 3+ 1.3 2.4.5).( 12 )5+1.3.52.4.6.7( 1 2)+...) has been attributd to Newton in the late 16OO. The proof of this result uses the Madautin series for f(x)=sin1x . a. Prove that the series converges. b. Evaluate the partial sums S for n = 5. 10. 20. C. Compare S to z for n = 5. 10. 20 and discuss the number of correct decimal places.Series Converging to and 1/ Dozens of series exist that converge to or an algebraic expression containing . Here we look at several examples and compare their rates of convergence. By rate of conveigence. we mean the number of terms necessazy for a partial sum to be within a certain amount of the actual value. The series representations of z in the first two examples can be explained using Maclaunn series, which axe discussed in the next chapter. The third example relies on material beyond the scope of this text. 3. The series 1=89801n=0(4n)!(1103+26390n)( n!)43964n was discovered by Ramanujan in the early 1900s. William Gosper. Jr., used this series to calculate x to an accuracy of more than 17 million digits in the niid-l9)s. At the time, that was a world record. Since that time, this series and others by Rarnanujan have led mathematicians to find many other series represetnations for x and I/x. a. Prove that this series converges. b. Evaluate the first term in this series. Compare this number with the value of x from a calculating utilky. To how many decimal places do these two numbers agree? What if we add the first two teiTns in the series? C. Investigate the life of Szinivasa Ram.anujan (1887—1920) and write a brief summary. Ramanujan is one of the most fascinating stories in the histo, of mathematics. He was basically sell-taught, with no formal training in mathematics. e he contributed in highly original ways to many advanced areas of mathemarics.Use the ratio test to determine whether n=1anconverges, where anis given in the following problems. State if the ratio test is inconclusive. 317. an=1/n!Use the ratio test to determine whether n=1anconverges, where anis given in the following problems. State if the ratio test is inconclusive. 318. an=10n/n!Use the ratio test to determine whether n=1anconverges, where anis given in the following problems. State if the ratio test is inconclusive. 319. an=n2/2n Use the ratio test to determine whether n=1anconverges, where anis given in the following problems. State if the ratio test is inconclusive. 320. an=n10/2n Use the ratio test to determine whether n=1anconverges, where anis given in the following problems. State if the ratio test is inconclusive. 321. n=1( n!)3(3n!)Use the ratio test to determine whether n=1anconverges, where anis given in the following problems. State if the ratio test is inconclusive. 322. n=123n( n!)3(3n!) .Use the ratio test to determine whether n=1anconverges, where anis given in the following problems. State if the ratio test is inconclusive. 323. n=123n( n!)3n2n Use the ratio test to determine whether n=1anconverges, where anis given in the following problems. State if the ratio test is inconclusive. 324. n=1(2n)!( 2n)n Use the ratio test to determine whether n=1anconverges, where anis given in the following problems. State if the ratio test is inconclusive. 325. n=1n!( n/e)n Use the ratio test to determine whether n=1anconverges, where anis given in the following problems. State if the ratio test is inconclusive. 326. n=1(2n)( n/e)2n Use the ratio test to determine whether n=1anconverges, where anis given in the following problems. State if the ratio test is inconclusive. 327. n=1( 2 nn)2( 2n)2n Use the root test to determine whether n=1an converges, where an, is as follows. 328. ak=(k12k+3)k Use the root test to determine whether n=1an converges, where an, is as follows. 330. an=(lnn)2nnn Use the root test to determine whether n=1an converges, where an, is as follows. 331. an=n/en Use the root test to determine whether n=1an converges, where an, is as follows. 332. an=n/en Use the root test to determine whether n=1an converges, where an, is as follows. 333. ak=keek Use the root test to determine whether n=1an converges, where an, is as follows. 334. ak=kk Use the root test to determine whether n=1an converges, where an, is as follows. 335. an=(1e+1n)n Use the root test to determine whether n=1an converges, where an, is as follows. 336. ak=1(1+lnk)k Use the root test to determine whether n=1an converges, where an, is as follows. 337. an=(ln( 1+lnn))n(lnn)n In the following exercises, use either the ratio test or the root test as appropriate to determine whether the series k=1ak with given terms ak converges, or state if the test is inconclusive. 338. ak=k!1.3.5...(2k1)In the following exercises, use either the ratio test or the root test as appropriate to determine whether the series k=1ak with given terms ak converges, or state if the test is inconclusive. 339. ak=1.4.7...(3k2)3kk!In the following exercises, use either the ratio test or the root test as appropriate to determine whether the series k=1ak with given terms ak converges, or state if the test is inconclusive. 340. ak=1.4.7...(3k2)3kk!In the following exercises, use either the ratio test or the root test as appropriate to determine whether the series k=1ak with given terms ak converges, or state if the test is inconclusive. 341. an=(11n)n2 In the following exercises, use either the ratio test or the root test as appropriate to determine whether the series k=1ak with given terms ak converges, or state if the test is inconclusive. 342. ak=(1k+1+1k+2+...+12k)k In the following exercises, use either the ratio test or the root test as appropriate to determine whether the series k=1ak with given terms ak converges, or state if the test is inconclusive. 343. ak=(1k+1+1k+2+...+13k)k In the following exercises, use either the ratio test or the root test as appropriate to determine whether the series k=1ak with given terms ak converges, or state if the test is inconclusive. 344. an=(n1/n1)n Use the ratio test to determine whether n=1ana converges, or state if the ratio test is inconclusive. 345. n=13n22 n 3 Use the ratio test to determine whether n=1ana converges, or state if the ratio test is inconclusive. 346. n=12n2nnn!Use the root and limit comparison tests to determine whether n=1an converges. 347. an=1/xnnwhere xn+1=12xn+1xn,x1=1 (Hint: Find limit of xn.)In the following exercises, use an appropriate test to determine whether the series converges. 348. d=1(n+1)n3+n2+n+1 In the following exercises, use an appropriate test to determine whether the series converges. 349. d=1( 1)n+1(n+1)n3+3n2+3n+1 In the following exercises, use an appropriate test to determine whether the series converges. 350. n=1( n+1)2n3+( 1,1)n In the following exercises, use an appropriate test to determine whether the series converges. 351. n=1( n1)n( n+1)n In the following exercises, use an appropriate test to determine whether the series converges. 352. an=(1+1 n 2)n(Hint:(1+ 1 n 2)n2e.)In the following exercises, use an appropriate test to determine whether the series converges. 353. ak=1/2sin2k In the following exercises, use an appropriate test to determine whether the series converges. 354. ak=2sin(1/k)In the following exercises, use an appropriate test to determine whether the series converges. 355. an=1/(nn+2)where(kn)=n!k!(nk)!In the following exercises, use an appropriate test to determine whether the series converges. 356. ak=1/(k2k)In the following exercises, use an appropriate test to determine whether the series converges. 357. ak=2k/(k3k)In the following exercises, use an appropriate test to determine whether the series converges. 358. ak=(k k+lnk)kak=(1+ lnkk)(k/lnk)lnkelnk.)In the following exercises, use an appropriate test to determine whether the series converges. 359. ak=(kk+lnk)2k(Hint:ak=(1+lnkk)(k/ln)lnk2 The following series converge by the ratio test. Use summation by parts, k=1nak(bk+1bk)=[an+1bn+1a1b1]k=1nbk+1( a k+1 a k) to find the sum of the given series. 360. k=1k2k (Hint: Take ak=k and bk=21k .)The following series converge by the ratio test. Use summation by parts, k=1nak(bk+1bk)=[an+1bn+1a1b1]k=1nbk+1( a k+1 a k) to find the sum of the given series. 361. k=1kck, where c>1 (Hint: Take ak=k and bk=c1k/(c1).)The following series converge by the ratio test. Use summation by parts, k=1nak(bk+1bk)=[an+1bn+1a1b1]k=1nbk+1( a k+1 a k) to find the sum of the given series. 362. n=1n22n The following series converge by the ratio test. Use summation by parts, k=1nak(bk+1bk)=[an+1bn+1a1b1]k=1nbk+1( a k+1 a k) to find the sum of the given series. 363. n=1( n+1)22n The kth term of each of the following series has a factor x. Find the range of x for which the ratio test implies that the series converges. 364. k=1xkk2 The kth term of each of the following series has a factor x. Find the range of x for which the ratio test implies that the series converges. 365. k=1x2k3k The kth term of each of the following series has a factor x. Find the range of x for which the ratio test implies that the series converges. 366. k=1x2k3k The kth term of each of the following series has a factor x. Find the range of x for which the ratio test implies that the series converges. 367. k=1xkk!Does there exist a number p such that n=12nnp. converges?Let 0 < r < 1. For which real numbers p does n=1nprnconverge?Suppose that limn|an+1an|=p . For which values of p must n=12nan converge?Suppose that limn|an+1an|=p . For which values of r>0 is n=1rnanguaranteed to converge?Suppose that |an+1an|(n+1)p for all n = 1. 2,... where p is a fixed real number. For which values of p is n=1n!an a guaranteed to converge?For which values of r>0. if any, does n=1rn converge? (Him: n=1an=k=1 n=k2 ( k+1 )2 1 a n)Suppose that |an+2a2|r1 for all n. Can you conclude that n=1an converges?Let an=2[n/2] where [x] is the greatest integer less than or equal to x. Determine whether n=1an converges and justify your answer. The following advanced exercises use a generalized ratio test to determine convergence of some series that arise in particular applications when tests in this chapter, including the ratio and root test, are not powerful enough to determine their convergence. The test states that if limna2nan1/2 . then an converges, while if limna2n+1an1/2 then a diverges.Let an=143658...2n12n+2=1.3.5...(2n1)2n(n+1)! Explain why the ratio test cannot determine convergence of n=1an. Use the fact that 1 —1/(4k) is increasing k to a estimate limna2nan,Let an=11+x22+x...nn+x1n=(n1)!(1+x)(2+x)...(n+x). Show that a2n/anex/2/2. For which x>0 does the generalized ratio test imply convergence of n=1an(Hint: Write 2a2n/an as a product of n factors each smaller than 1/(1 + x/(2n)).)Letan=nlnn(lnn)n,Showthata2nan0asn.True or False? Justify your answer with a proof or a counterexample. 379. If limnan=0, then n=1an converges.True or False? Justify your answer with a proof or a counterexample. 380. If limnan=0 , then n=1an diverges.True or False? Justify your answer with a proof or a counterexample. 381. If n=1an converges, then n=1an converges True or False? Justify your answer with a proof or a counterexample. 382. If n=12nan converges, then n=1(2)nan converges.Is the sequence bounded, monotone, and convergent or divergent? If it is convergent, find the limit. 383. an=3+n21n Is the sequence bounded, monotone, and convergent or divergent? If it is convergent, find the limit. 384. an=ln(1n)Is the sequence bounded, monotone, and convergent or divergent? If it is convergent, find the limit. 385. an=ln(n+1)n+1 Is the sequence bounded, monotone, and convergent or divergent? If it is convergent, find the limit. 386. an=2n+15n Is the sequence bounded, monotone, and convergent or divergent? If it is convergent, find the limit. 387. an=ln(cosn)n Is the series convergent or divergent? 388. n=11n2+5n+4 Is the series convergent or divergent? 389. n=1ln(n+1n)Is the series convergent or divergent? 390. n=1ln2nn4 Is the series convergent or divergent? 391. n=1enn!Is the series convergent or divergent? 392. n=1n(n+1/n)Is the series convergent or divergent? If convergent, is it absolutely convergent? 393. n=1( 1)nn Is the series convergent or divergent? If convergent, is it absolutely convergent? 394. n=1( 1)nn!3n Is the series convergent or divergent? If convergent, is it absolutely convergent? 395. n=1( 1)nn!nn Is the series convergent or divergent? If convergent, is it absolutely convergent? 396. n=1sin(n2)Is the series convergent or divergent? If convergent, is it absolutely convergent? 397. n=1cos(n)en Evaluate 398. n=12n+47n Evaluate 399. n=11(n+1)(n+2)A legend from India tells that a mathematician invented chess for a king. The king enjoyed the game so much lie allowed the mathematician to demand any payment. The mathematician asked for one grain of rice for the first square on the chessboard, two grains of rice for the second square on the chessboard, and so on. Find an exact expression for the total payment (in grains of rice) requested by the mathematician. Assuming there are 30,000 grains of rice in 1 pound, and 2000 pounds in 1 ton, how many tons of rice did the mathematician attempt to receive?The following problems consider a simple population model of the housefly, which can be exhibited by the recursive formula xn+1=bxn. where xnis the population of houseflies at generation n. and b is the average number of offspring per housefly who survive to the next generation. Assume a starting population x0. 401. Find limnxn if b > 1, b < 1, and b=1.The following problems consider a simple population model of the housefly, which can be exhibited by the recursive formula xn+1=bxn. where xnis the population of houseflies at generation n. and b is the average number of offspring per housefly who survive to the next generation. Assume a starting population x0. 402. Find an expression for Sn=i=0nxi in terms of b and x0. What does it physically represent?The following problems consider a simple population model of the housefly, which can be exhibited by the recursive formula xn+1=bxn. where xnis the population of houseflies at generation n. and b is the average number of offspring per housefly who survive to the next generation. Assume a starting population x0. 403. If b=34 and x0=100 , find S10 and limnSn The following problems consider a simple population model of the housefly, which can be exhibited by the recursive formula xn+1=bxn. where xnis the population of houseflies at generation n. and b is the average number of offspring per housefly who survive to the next generation. Assume a starting population x0. 404. For what values of b will the series converge and diverge? What does the series converge to?In the following exercises, state whether each statement is true, or give an example to show that Ii is false. 1. If n=1anxn converges, then anxn0 as n .In the following exercises, state whether each statement is true, or give an example to show that Ii is false. 2. n=1anxn converges at x = 0 for any real numbers xn In the following exercises, state whether each statement is true, or give an example to show that Ii is false. 3. Given any sequence a. there is always some R >0. possibly very small, such that a, n=1anxn converges on (R, R).In the following exercises, state whether each statement is true, or give an example to show that Ii is false. 4. If n=1anxnhas radius of convergence R >0 and if |bn||an| if for all n, then the radius of convergence of n=1bnxnis greater than or equal to R.In the following exercises, state whether each statement is true, or give an example to show that Ii is false. 5. Suppose that n=1an(x3)n converges at x = 6. At which of the following points must the series also converge? Use the fact that if an(xc)n conveiges at x. then ii converges at any point closer to c than x. a. x=1 b. x=2 c. x = 3 d. x = 0 e. x =5.99 f. x = 0.000001 In the following exercises, state whether each statement is true, or give an example to show that Ii is false. 6. Suppose that n=0an(x+1)n converges at x=-2 . At which of the following points must die series also converge? Use the fact that if an(xc)nconverges at x, then t converges at any point closer to c than x. a. x = 2 b. x = -1 c. x = -3 d. x = 0 e. x = 0.99 f. x = 0.000001 In the following exercises, suppose that |an+1an|1 as n. Find the radius of convergence for each series. 7. n=0an2nxn In the following exercises, suppose that |an+1an|1 as n. Find the radius of convergence for each series. 8. n=0anxn2n In the following exercises, suppose that |an+1an|1 as n. Find the radius of convergence for each series. 9. n=0annxnen In the following exercises, suppose that |an+1an|1 as n. Find the radius of convergence for each series. 10. n=0 an (1)n xn 10n In the following exercises, suppose that |an+1an|1 as n. Find the radius of convergence for each series. 11. n=0an(1)nx2n In the following exercises, suppose that |an+1an|1 as n. Find the radius of convergence for each series. 12. n=0an(4)nx2n In the following exercises, find the radius of convergence R and interval of convergence for anxn with the given coefficients an. 13. n=1 (2x)nn In the following exercises, find the radius of convergence R and interval of convergence for anxn with the given coefficients an. 14. n=1(1)nxnn In the following exercises, find the radius of convergence R and interval of convergence for anxn with the given coefficients an. 15. n=1nxn2n In the following exercises, find the radius of convergence R and interval of convergence for anxn with the given coefficients an. 16. n=1nxnen In the following exercises, find the radius of convergence R and interval of convergence for anxn with the given coefficients an. 17. n=1n2x22n In the following exercises, find the radius of convergence R and interval of convergence for anxn with the given coefficients an. 18. k=1kexkek In the following exercises, find the radius of convergence R and interval of convergence for anxn with the given coefficients an. 19. k=1kxkk In the following exercises, find the radius of convergence R and interval of convergence for anxn with the given coefficients an. 20. n=1xnn!In the following exercises, find the radius of convergence R and interval of convergence for anxn with the given coefficients an. 21. n=1 10nxnn!In the following exercises, find the radius of convergence R and interval of convergence for anxn with the given coefficients an. 22. n=1(1)nxn1n(2n)In the following exercises, find the radius of convergence of each series. 23. k=1 (k!)2xk(2k)!In the following exercises, find the radius of convergence of each series. 24. n=1(2n)!xnn 2n In the following exercises, find the radius of convergence of each series. 25. k=1k!135(2k1)xk In the following exercises, find the radius of convergence of each series. 26. k=12462k(2k)!xn In the following exercises, find the radius of convergence of each series. 27. n=1xn( 2n n )where(nk)=n!k!(nk)!In the following exercises, find the radius of convergence of each series. 28. n=1sin2nxn In the following exercises, use the ratio test to determine the radius of convergence of each series. 29. 29.n=1 (n!)3(3n)!xn

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