Consider the following statements. (i) 2 (-1)"Cn must converge if cn > Cn+1 > 0. Сп n =0 1 (ii) 2 bn must converge if 0 < bn < 213 . n=1

Q: (4x+8)* What is the radius of convergence ( R) and center (a ) of the power series E 3n

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Q: Determine of each point convergens or diverges: (-1)"(31)/n! (n + 1)! (1)E (n+ 2)! n=0

A: As per the guidelines, I will solve only the first question. According to the limit comparison test,…

Q: 2. Assume that p > 1. (a) For all n> 1 prove that P-1 (n + 1)P (b) Prove that is convergent. NP <…

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Q: 55 n=2 1 13/2 ไn n

A: Given that, the series is           Let  , then .

Q: D) { (-)"} [Select] [Select] converges to 2.

A: Please see the below picture for detailed solution.

Q: Which of the followings are true? 1. E is absolutely convergent. 00 n=1 n! II. En-1 (-1)-1 is…

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Q: Q4. Either find the smallest positive integer solution to x² - 29y² = 1, or compute the 9-th…

A: We need to find the smallest positive integer solution to  or compute the  convergent to .We will…

Q: 3. Assume the sequence of positive numbers (sn) increases monotonically to infinity. Prove that Sn+1…

A: Let Sn be a sequence of positive numbers sn increase monotonically to infinity. ⇒limn→∞sn=∞     ∵ sn…

Q: (1) E(-1)n+1 (3n)! 3n+1 n=1

A: (f). Given series is ∑n=1+∞(-1)n+1(3n)!3n+1

Q: 2 then E " Cn+1 converges. 3.11 limnr+m

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Q: (-1)"(x + 1)" 5" 7 Problem 1: The power series has radius of convergence R= 5. What is n=1 the…

A: Given:∑n=1∞-1nx+1n5nnUsing ratio testlimn→∞an+1an&lt;1

Q: Hint: Analytical answer is required to be uploaded in Moodle 1. Consider the differential equation…

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Q: (3) Find all p ER so that 3n² + 2n +1 n2p-1 + 137 n=1 converges. As always, your answer must be…

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Q: ∞ 2. Does -1 4n converge? Why n! or why not?

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Q: Which of the followings are true? (-1)" 1. En-1 n²+Inn is absolutely convergent. (-1)* II. En-1 is…

A: We have to find the option

Q: Suppose that xo 22 and xn = 2 + √xn-1-2 for ne N. Use the Monotone Convergence Theorem to show that…

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Q: a) For which values of x does f(x) =E n=0 converge and for which values ofx does it diverge? b) When…

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Q: (b) Determine whether Σ n=1 ditionally, or diverges. (-1)n.n²+3 n²-4 converges absolutely, converges…

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Q: 2. Consider the function f(x)=x²+5 on the interval [1,3]. If the interval [1,3] is divided into a…

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Q: f(x) = 6 x²-2x-8 f(x)=- -((())) (-1)"+1 4"+1 2+1 Σ Find the interval of convergence. (Enter your…

A: fx=∑n=0∞-14n+1+-1n+12n+1xn

Q: 1 Σ 8n2 – 16n + 6 n=1

A: We will use limit form of comparison test to determine whether given series is convergent or…

Q: Prove the following sequences diverge. (-5)"т + 3 2 sin ( 10"T + 4 en +n? +n +11 (Ъ) » (* ) 2n n -

A: We use basic of sequence.

Q: For eksample Xn = n² (n+1)! investigate whether the serles Σ 8 n=1 Xn convergent Or divergent. Prove…

A: We will use ratio test to check whether given series is convergent or not.

Q: Show by example that Σ(an/bn) may converge to something other than A/B even when A = gan, B = Σbn ≠…

A: Given that an, bn≠0 Let us consider ∑an converges to A. Also ∑bn converges to B. To show that ∑anbn…

Q: Which of the followings are false? is absolutely convergent. 1. Ln=1 (-1)*-1 is absolutely…

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Q: Suppose (xn) does not converge to 0. Then there exists e>0 such that all except for finitely many…

A: Consider the sequence xn=1,0,1,0,1,0,....... Here the odd terms are 1 and the even terms are 0.…

Q: 7) The point p = 3 is a zero of the function f(x) = x³ – 7x² + 15x – 9, using Newton iteration to…

A: Given that fx=x3-7x2+15x-9. 3 is a root of this function. To find f'x: f'x=3x2-14x+15 At x=3,…

Q: Σ Suppose that ax" converges to a function y such that y" – 2y' + y = 0 where y(0) = 0 and y'(0) =…

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Q: Prove the first part of FACT 2, i.e., if an converges and Σ bn diverges, then Σ(an + bn) diverges.…

A: Let Σan converges and Σbn diverges then we have to prove Σ(an+bn) diverges

Q: For each of (a)-(d) give a sequence (rn)1 with the stated prop- erties. In each case, the sequence…

A: As per our guidelines we are suppose to answer only 3 sub-parts.

Q: Consider the equation 2x - 1 = 2 on the interval [1, 2]. 1 +1 will converge to a 2pn-1 (a) Show that…

A: * SOLUTION :-   Based on the above information the calculation is given below.

Q: 2. (a) Write the first 5 terms of the sequences with general terms an below and indicate whether…

A: As per the guidelines, we are suppose to answer only the first 3 subparts.  Repost the rest of the…

Q: Suppose that ∞ n = 0 anxn converges to a function y such that y'' − 2y' + y = 0 where y(0) = 0 and…

A: We need to find formula.

Q: 16.) Given a, >0 for all k and given a, is convergent, then which one of the following must also k-1…

A: Q. 16 Given ak&gt;0 for all k and the series ∑k=1∞ak is convergent. Which means ∑k=1∞ak = Some…

Q: Sn/2", l1/2". if n is a prime number otherwise. Let an Does Ea, converge? Give reasons for your…

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Q: 2. Give examples (without proofs) of the following: a) sequences {xn} and {yn} which both diverge,…

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Q: 00 7" + 2" Determine whether the series converges or diverges. If it converges, find its sum. 9h n=…

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Q: Which of the following statement(s) is/are true? 1 n+9 00 « Σ (-1)* converges conditionally. n=1 80…

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Q: 00 (-1)*+1 7n2 + 2 n=1 Input C for convergence and D for divergence:

A: Here,