Calculus: Early Transcendentals
Calculus: Early Transcendentals
9th Edition
ISBN: 9780357631478
Author: James Stewart, Daniel K. Clegg, Saleem Watson
Publisher: Cengage Learning
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Chapter 11.6, Problem 46E

Around 1910, the Indian mathematician Srinivasa Ramanujan discovered the formula

1 π = 2 2 9801 ∑ n = 0 ∞ ( 4 n ) ! ( 1103 + 26390 n ) ( n ! ) 4 396 4 n

William Gosper used this series in 1985 to compute the first 17 million digits of π.

(a) Verify that the series is convergent.

(b) How many correct decimal places of π do you get if you use just the first term of the series? What if you use two terms?

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Chapter 11 Solutions

Calculus: Early Transcendentals

Ch. 11.1 - (a) What is a sequence? (b) What does it mean to...Ch. 11.1 - (a) What is a convergent sequence? Give two...Ch. 11.1 - List the first five terms of the sequence. 3....Ch. 11.1 - List the first five terms of the sequence. 4....Ch. 11.1 - List the first five terms of the sequence. 5....Ch. 11.1 - List the first five terms of the sequence. 6....Ch. 11.1 - List the first five terms of the sequence. 7....Ch. 11.1 - List the first five terms of the sequence. 8....Ch. 11.1 - List the first five terms of the sequence. 9....Ch. 11.1 - List the first five terms of the sequence. 10....
Ch. 11.1 - List the first five terms of the sequence. 11....Ch. 11.1 - List the first five terms of the sequence. 12....Ch. 11.1 - List the first five terms of the sequence. 13....Ch. 11.1 - List the first five terms of the sequence. 10. a1...Ch. 11.1 - List the first five terms of the sequence. 11. a1...Ch. 11.1 - List the first five terms of the sequence. 12. a1...Ch. 11.1 - Find a formula for the general term an of the...Ch. 11.1 - Find a formula for the general term an of the...Ch. 11.1 - Find a formula for the general term an of the...Ch. 11.1 - Find a formula for the general term an of the...Ch. 11.1 - Find a formula for the general term an of the...Ch. 11.1 - Find a formula for the general term an of the...Ch. 11.1 - Calculate, to four decimal places, the first ten...Ch. 11.1 - Calculate, to four decimal places, the first ten...Ch. 11.1 - Calculate, to four decimal places, the first ten...Ch. 11.1 - Calculate, to four decimal places, the first ten...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Determine whether sequence converges or diverges....Ch. 11.1 - Use a graph of the sequence to decide whether the...Ch. 11.1 - Prob. 64ECh. 11.1 - Use a graph of the sequence to decide whether the...Ch. 11.1 - Use a graph of the sequence to decide whether the...Ch. 11.1 - Prob. 67ECh. 11.1 - Use a graph of the sequence to decide whether the...Ch. 11.1 - Prob. 69ECh. 11.1 - (a) Determine whether the sequence defined as...Ch. 11.1 - If 1000 is invested at 6% interest, compounded...Ch. 11.1 - If you deposit 100 at the end of every month into...Ch. 11.1 - Prob. 73ECh. 11.1 - Prob. 74ECh. 11.1 - For what values of r is the sequence {nrn}...Ch. 11.1 - (a) If {an} is convergent, show that...Ch. 11.1 - Suppose you know that {an} is a decreasing...Ch. 11.1 - Determine whether the sequence is increasing,...Ch. 11.1 - Determine whether the sequence is increasing,...Ch. 11.1 - Determine whether the sequence is increasing,...Ch. 11.1 - Determine whether the sequence is increasing,...Ch. 11.1 - Determine whether the sequence is increasing,...Ch. 11.1 - Determine whether the sequence is increasing,...Ch. 11.1 - Determine whether the sequence is increasing,...Ch. 11.1 - Find the limit of the sequence {2,22,222,...}Ch. 11.1 - A sequence {an} is given by a1=2,an+1=2+an. (a) By...Ch. 11.1 - Show that the sequence defined by a1=1an+1=31an is...Ch. 11.1 - Show that the sequence defined by a1=2an+1=13an...Ch. 11.1 - (a) Fibonacci posed the following problem: Suppose...Ch. 11.1 - (a) Let a1 = a, a2 = f(a), a3 = f(a2) = f(f(a)), ....Ch. 11.1 - (a) Use a graph to guess the value of the limit...Ch. 11.1 - Use Definition 2 directly to prove that limn rn =...Ch. 11.1 - Prove Theorem 6. [Hint: Use either Definition 2 or...Ch. 11.1 - Use a graph of the sequence to decide whether the...Ch. 11.1 - Prove that if limn an = 0 and {bn} is bounded,...Ch. 11.1 - Let an=(1+1n)n. (a) Show that if 0 a b, then...Ch. 11.1 - Let a and b be positive numbers with a b. Let a1...Ch. 11.1 - (a) Show that if limn a2n = L and limn a2n+1 = L,...Ch. 11.1 - The size of an undisturbed fish population has...Ch. 11.1 - Prob. 1DPCh. 11.1 - Prob. 2DPCh. 11.1 - Prob. 3DPCh. 11.2 - (a) What is the difference between a sequence and...Ch. 11.2 - Explain what it means to say that n=1an=5.Ch. 11.2 - Calculate the sum of the seriesn=1an whose partial...Ch. 11.2 - Calculate the sum of the seriesn=1an whose partial...Ch. 11.2 - Prob. 5ECh. 11.2 - Calculate the first eight terms of the sequence of...Ch. 11.2 - Calculate the first eight terms of the sequence of...Ch. 11.2 - Calculate the first eight terms of the sequence of...Ch. 11.2 - Prob. 9ECh. 11.2 - Prob. 10ECh. 11.2 - Find at least 10 partial sums of the series. Graph...Ch. 11.2 - Find at least 10 partial sums of the series. Graph...Ch. 11.2 - Find at least 10 partial sums of the series. Graph...Ch. 11.2 - Let an=2n3n+1 (a) Determine whether {an} is...Ch. 11.2 - (a) Explain the difference between i=1naiandj=1naj...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the geometric series is...Ch. 11.2 - Determine whether the geometric series is...Ch. 11.2 - Determine whether the geometric series is...Ch. 11.2 - Determine whether the geometric series is...Ch. 11.2 - Determine whether the geometric series is...Ch. 11.2 - Determine whether the geometric series is...Ch. 11.2 - Determine whether the geometric series is...Ch. 11.2 - Determine whether the geometric series is...Ch. 11.2 - Determine whether the geometric series is...Ch. 11.2 - Determine whether the geometric series is...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Prob. 34ECh. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Prob. 48ECh. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Prob. 50ECh. 11.2 - Let X = 0.99999 .... (a) Do you think that x 1 or...Ch. 11.2 - Prob. 52ECh. 11.2 - Express the number as a ratio of integers. 51....Ch. 11.2 - Express the number as a ratio of integers. 52....Ch. 11.2 - Express the number as a ratio of integers. 53....Ch. 11.2 - Prob. 56ECh. 11.2 - Express the number as a ratio of integers. 55....Ch. 11.2 - Prob. 58ECh. 11.2 - Find the values of x for which the series...Ch. 11.2 - Find the values of x for which the series...Ch. 11.2 - Find the values of x for which the series...Ch. 11.2 - Find the values of x for which the series...Ch. 11.2 - Find the values of x for which the series...Ch. 11.2 - Prob. 64ECh. 11.2 - Find the values of x for which the series...Ch. 11.2 - Find the values of x for which the series...Ch. 11.2 - Prob. 67ECh. 11.2 - Use the partial fraction command on a computer...Ch. 11.2 - If the nth partial sum of a series n=1an is...Ch. 11.2 - If the nth partial sum of a series n=1an is sn = 3...Ch. 11.2 - A doctor prescribes a 100-mg antibiotic tablet to...Ch. 11.2 - A patient is injected with a drug every 12 hours....Ch. 11.2 - A patient takes 150 mg of a drug at the same time...Ch. 11.2 - After injection of a dose D of insulin, the...Ch. 11.2 - When money is spent on goods and services, those...Ch. 11.2 - A certain ball has the property that each time it...Ch. 11.2 - Find the value of c if n=2(1+c)n=2Ch. 11.2 - Find the value of c such that n=0enc=10Ch. 11.2 - Graph the curves y = xn, 0 x 1, for n = 0, 1, 2,...Ch. 11.2 - The figure shows two circles C and D of radius 1...Ch. 11.2 - Prob. 84ECh. 11.2 - What is wrong with the following calculation?...Ch. 11.2 - Prob. 86ECh. 11.2 - If an is divergent and c 0, show that can, is...Ch. 11.2 - Prob. 89ECh. 11.2 - Prob. 90ECh. 11.2 - Prob. 91ECh. 11.2 - The Fibonacci sequence was defined in Section 11.1...Ch. 11.2 - The Cantor set, named after the German...Ch. 11.2 - (a) A sequence an is defined recursively by the...Ch. 11.2 - Consider the series n=1n/(n+1)!. (a) Find the...Ch. 11.2 - In the figure at the right there are infinitely...Ch. 11.3 - Draw a picture to show that n=21n1.311x1.3dx What...Ch. 11.3 - Suppose f is a continuous positive decreasing...Ch. 11.3 - Use the Integral Test to determine whether the...Ch. 11.3 - Use the Integral Test to determine whether the...Ch. 11.3 - Use the Integral Test to determine whether the...Ch. 11.3 - Use the Integral Test to determine whether the...Ch. 11.3 - Use the Integral Test to determine whether the...Ch. 11.3 - Use the Integral Test to determine whether the...Ch. 11.3 - Use the Integral Test to determine whether the...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Explain why the Integral Test can't be used to...Ch. 11.3 - Explain why the Integral Test can't be used to...Ch. 11.3 - Find the values of p for which the series is...Ch. 11.3 - Find the values of p for which the series is...Ch. 11.3 - Find the values of p for which the series is...Ch. 11.3 - Find the values of p for which the series is...Ch. 11.3 - (a) Find the partial sum s10 of the series n=11/n4...Ch. 11.3 - (a) Use the sum of the first 10 terms to estimate...Ch. 11.3 - Find the sum of the series n=1ne2n correct to four...Ch. 11.3 - Estimate n=1(2n+1)6 correct to five decimal...Ch. 11.3 - How many terms of the series n=11[n(lnn)2] would...Ch. 11.3 - Prob. 43ECh. 11.3 - Prob. 44ECh. 11.3 - Prob. 45ECh. 11.3 - Find all positive values of b for which the series...Ch. 11.3 - Find all values of c for which the following...Ch. 11.4 - Suppose an and bn are series with positive terms...Ch. 11.4 - Suppose an and bn are series with positive terms...Ch. 11.4 - (a) Use the Direct Comparison Test to show that...Ch. 11.4 - (a) Use the Direct Comparison Test to show that...Ch. 11.4 - Which of the following inequalities can be used to...Ch. 11.4 - Which of the following inequalities can be used to...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Use the sum of the first 10 terms to approximate...Ch. 11.4 - Use the sum of the first 10 terms to approximate...Ch. 11.4 - Use the sum of the first 10 terms to approximate...Ch. 11.4 - Use the sum of the first 10 terms to approximate...Ch. 11.4 - The meaning of the decimal representation of a...Ch. 11.4 - For what values of p does the series n=21/(nPlnn)...Ch. 11.4 - Show that n=21(lnn)lnlnn diverges. [Hint: Use...Ch. 11.4 - (a) Suppose that an and bn are series with...Ch. 11.4 - (a) Suppose that an and bn are series with...Ch. 11.4 - Give an example of a pair of series an and bn with...Ch. 11.4 - Show that if an 0 and limnnan0 then an is...Ch. 11.4 - Show that if an 0 and an is convergent, then...Ch. 11.4 - If an is a convergent series with positive terms,...Ch. 11.4 - Prove that if an0 and an converges, then an2 also...Ch. 11.4 - Prob. 55ECh. 11.5 - (a) What is an alternating series? (b) Under what...Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence. 12....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - (a) What does it mean for a series to be...Ch. 11.5 - Determine whether the series is absolutely...Ch. 11.5 - Determine whether the series is absolutely...Ch. 11.5 - Determine whether the series is absolutely...Ch. 11.5 - Graph both the sequence of terms and the sequence...Ch. 11.5 - Graph both the sequence of terms and the sequence...Ch. 11.5 - Show that the series is convergent. How many terms...Ch. 11.5 - Show that the series is convergent. How many terms...Ch. 11.5 - Show that the series is convergent. How many terms...Ch. 11.5 - Show that the series is convergent. How many terms...Ch. 11.5 - Approximate the sum of the series correct to four...Ch. 11.5 - Approximate the sum of the series correct to four...Ch. 11.5 - Prob. 43ECh. 11.5 - Prob. 44ECh. 11.5 - Is the 50th partial sum s50 of the alternating...Ch. 11.5 - For what values of p is each series convergent?...Ch. 11.5 - For what values of p is each series convergent?...Ch. 11.5 - For what values of p is each series convergent?...Ch. 11.5 - Prob. 49ECh. 11.5 - Prob. 50ECh. 11.6 - What can you say about the series an in each of...Ch. 11.6 - Determine whether the series is absolutely...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Root Test to determine whether the series...Ch. 11.6 - Use the Root Test to determine whether the series...Ch. 11.6 - Use the Root Test to determine whether the series...Ch. 11.6 - Use the Root Test to determine whether the series...Ch. 11.6 - Use the Root Test to determine whether the series...Ch. 11.6 - Use the Root Test to determine whether the series...Ch. 11.6 - Use any test to determine whether the series is...Ch. 11.6 - Use any test to determine whether the series is...Ch. 11.6 - Use any test to determine whether the series is...Ch. 11.6 - Use any test to determine whether the series is...Ch. 11.6 - Use any test to determine whether the series is...Ch. 11.6 - Use any test to determine whether the series is...Ch. 11.6 - Use any test to determine whether the series is...Ch. 11.6 - Use any test to determine whether the series is...Ch. 11.6 - The terms of a series are defined recursively by...Ch. 11.6 - A series an, is defined by the equations...Ch. 11.6 - Let {bn} be a sequence of positive numbers that...Ch. 11.6 - Let {bn} be a sequence of positive numbers that...Ch. 11.6 - For which of the following series is the Ratio...Ch. 11.6 - For which positive integers k is the following...Ch. 11.6 - (a) Show that n=0xn/n! converges for all x. (b)...Ch. 11.6 - Prob. 42ECh. 11.6 - (a) Find the partial sum s5 of the series...Ch. 11.6 - Prove the Root Test. [Hint for part (i): Take any...Ch. 11.6 - Around 1910, the Indian mathematician Srinivasa...Ch. 11.7 - Test the series for convergence or divergence. 1....Ch. 11.7 - Test the series for convergence or divergence. 2....Ch. 11.7 - Test the series for convergence or divergence. 3....Ch. 11.7 - Test the series for convergence or divergence. 4....Ch. 11.7 - Test the series for convergence or divergence. 5....Ch. 11.7 - Test the series for convergence or divergence. 6....Ch. 11.7 - Test the series for convergence or divergence. 7....Ch. 11.7 - Test the series for convergence or divergence. 8....Ch. 11.7 - Test the series for convergence or divergence. 9....Ch. 11.7 - Test the series for convergence or divergence. 10....Ch. 11.7 - Test the series for convergence or divergence. 11....Ch. 11.7 - Test the series for convergence or divergence. 12....Ch. 11.7 - Test the series for convergence or divergence. 13....Ch. 11.7 - Test the series for convergence or divergence. 14....Ch. 11.7 - Test the series for convergence or divergence. 15....Ch. 11.7 - Test the series for convergence or divergence. 16....Ch. 11.7 - Test the series for convergence or divergence. 17....Ch. 11.7 - Test the series for convergence or divergence. 18....Ch. 11.7 - Test the series for convergence or divergence. 19....Ch. 11.7 - Test the series for convergence or divergence. 20....Ch. 11.7 - Test the series for convergence or divergence. 21....Ch. 11.7 - Test the series for convergence or divergence. 22....Ch. 11.7 - Test the series for convergence or divergence. 23....Ch. 11.7 - Test the series for convergence or divergence. 24....Ch. 11.7 - Test the series for convergence or divergence. 33....Ch. 11.7 - Test the series for convergence or divergence. 34....Ch. 11.7 - Test the series for convergence or divergence. 25....Ch. 11.7 - Test the series for convergence or divergence. 26....Ch. 11.7 - Test the series for convergence or divergence. 27....Ch. 11.7 - Test the series for convergence or divergence. 28....Ch. 11.7 - Test the series for convergence or divergence. 29....Ch. 11.7 - Test the series for convergence or divergence. 30....Ch. 11.7 - Test the series for convergence or divergence. 31....Ch. 11.7 - Test the series for convergence or divergence. 32....Ch. 11.7 - Test the series for convergence or divergence. 33....Ch. 11.7 - Test the series for convergence or divergence. 34....Ch. 11.7 - Test the series for convergence or divergence. 35....Ch. 11.7 - Test the series for convergence or divergence. 36....Ch. 11.7 - Test the series for convergence or divergence. 37....Ch. 11.7 - Test the series for convergence or divergence. 38....Ch. 11.8 - What is a power series?Ch. 11.8 - (a) What is the radius of convergence of a power...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Prob. 10ECh. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - If n=0cn4n is convergent, can we conclude that...Ch. 11.8 - Graph the first several partial sums sn(x) of the...Ch. 11.8 - A function f is defined by f(x)=1+2x+x2+2x3+x4+...Ch. 11.8 - If f(x)=n=0cnxn, where cn+4 = cn for all n 0,...Ch. 11.8 - Prob. 41ECh. 11.8 - Prob. 42ECh. 11.8 - Prob. 43ECh. 11.8 - Suppose that the power series cn(xa)n satisfies...Ch. 11.8 - Suppose the series cnxn has radius of convergence...Ch. 11.8 - Suppose that the radius of convergence of the...Ch. 11.9 - If the radius of convergence of the power series...Ch. 11.9 - Suppose you know that the series n=0bnxn converges...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Express the function as the sum of a power series...Ch. 11.9 - Express the function as the sum of a power series...Ch. 11.9 - (a) Use differentiation to find a power series...Ch. 11.9 - (a) Use Equation I to find a power series...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for f , and...Ch. 11.9 - Find a power series representation for f , and...Ch. 11.9 - Find a power series representation for f , and...Ch. 11.9 - Find a power series representation for f , and...Ch. 11.9 - Evaluate the indefinite integral as a power...Ch. 11.9 - Evaluate the indefinite integral as a power...Ch. 11.9 - Evaluate the indefinite integral as a power...Ch. 11.9 - Evaluate the indefinite integral as a power...Ch. 11.9 - Use a power series to approximate the definite...Ch. 11.9 - Use a power series to approximate the definite...Ch. 11.9 - Use a power series to approximate the definite...Ch. 11.9 - Use a power series to approximate the definite...Ch. 11.9 - Use the result of Example 7 to compute arctan 0.2...Ch. 11.9 - Prob. 36ECh. 11.9 - (a) Show that the function f(x)=n=0xnn! is a...Ch. 11.9 - Show that the function f(x)=n=0(1)nx2n(2n)! is a...Ch. 11.9 - (a) Show that J0 (the Bessel function of order 0...Ch. 11.9 - The Bessel function of order l is defined by...Ch. 11.9 - Prob. 41ECh. 11.9 - Prob. 42ECh. 11.9 - Prob. 43ECh. 11.9 - Let fn(x)=(sinnx)/n2.Show that the series fn(x)...Ch. 11.9 - Let f(x)=n=1xnn2 Find the intervals of convergence...Ch. 11.9 - (a) Starting with lhe geometric series n=0xn, find...Ch. 11.9 - Prob. 47ECh. 11.9 - Use the power series for tan-1x to prove the...Ch. 11.9 - (a) By completing the square, show that...Ch. 11.9 - Prob. 51ECh. 11.10 - If f(x)=n=0bn(x5)n for all x, write a formula for...Ch. 11.10 - Prob. 2ECh. 11.10 - If f(n)(0) = (n + 1)! for n = 0, 1, 2, , find the...Ch. 11.10 - Find the Taylor series for f centered at 4 if...Ch. 11.10 - Use the definition of a Taylor series to find the...Ch. 11.10 - Use the definition of a Taylor series to find the...Ch. 11.10 - Use the definition of a Taylor series to find the...Ch. 11.10 - Use the definition of a Taylor series to find the...Ch. 11.10 - Use the definition of a Taylor series to find the...Ch. 11.10 - Use the definition of a Taylor series to find the...Ch. 11.10 - Find the Maclaurin series for f(x) using the...Ch. 11.10 - Find the Maclaurin series for f(x) using the...Ch. 11.10 - Find the Maclaurin series for f(x) using the...Ch. 11.10 - Find the Maclaurin series for f(x) using the...Ch. 11.10 - Prob. 15ECh. 11.10 - Prob. 16ECh. 11.10 - Find the Maclaurin series for f(x) using the...Ch. 11.10 - Find the Maclaurin series for f(x) using the...Ch. 11.10 - Find the Maclaurin series for f(x) using the...Ch. 11.10 - Find the Maclaurin series for f(x) using the...Ch. 11.10 - Find the Taylor series for .f(x) centered at the...Ch. 11.10 - Find the Taylor series for .f(x) centered at the...Ch. 11.10 - Find the Taylor series for .f(x) centered at the...Ch. 11.10 - Find the Taylor series for .f(x) centered at the...Ch. 11.10 - Find the Taylor series for f(x) centered at the...Ch. 11.10 - Prob. 26ECh. 11.10 - Prob. 27ECh. 11.10 - Find the Taylor series for f(x) centered at the...Ch. 11.10 - Find the Taylor series for f(x) centered at the...Ch. 11.10 - Find the Taylor series for f(x) centered at the...Ch. 11.10 - Prob. 31ECh. 11.10 - Prob. 32ECh. 11.10 - Prob. 33ECh. 11.10 - Prob. 34ECh. 11.10 - Use the binomial series to expand the function as...Ch. 11.10 - Use the binomial series to expand the function as...Ch. 11.10 - Use the binomial series to expand the function as...Ch. 11.10 - Use the binomial series to expand the function as...Ch. 11.10 - Use a Maclaurin series in Table 1 to obtain the...Ch. 11.10 - Use a Maclaurin series in Table 1 to obtain the...Ch. 11.10 - Use a Maclaurin series in Table 1 to obtain the...Ch. 11.10 - Use a Maclaurin series in Table 1 to obtain the...Ch. 11.10 - Use a Maclaurin series in Table 1 to obtain the...Ch. 11.10 - Use a Maclaurin series in Table 1 to obtain the...Ch. 11.10 - Use a Maclaurin series in Table 1 to obtain the...Ch. 11.10 - Use a Maclaurin series in Table 1 to obtain the...Ch. 11.10 - Use a Maclaurin series in Table 1 to obtain the...Ch. 11.10 - Use a Maclaurin series in Table 1 to obtain the...Ch. 11.10 - Use the definitions sinhx=exex2coshx=ex+ex2 and...Ch. 11.10 - Use the formula tanh1x=12ln1+x1x1x1 and the...Ch. 11.10 - Find the Maclaurin series of f (by any method) and...Ch. 11.10 - Prob. 52ECh. 11.10 - Find the Maclaurin series of f (by any method) and...Ch. 11.10 - Find the Maclaurin series of f (by any method) and...Ch. 11.10 - Use the Maclaurin series for cos x to compute cos...Ch. 11.10 - Use the Maclaurin series for ex to calculate 1/e10...Ch. 11.10 - (a) Use the binomial series to expand 1/1x2. (b)...Ch. 11.10 - (a) Expand 1/1+x4 as a power series. (b) Use part...Ch. 11.10 - Evaluate the indefinite integral as an infinite...Ch. 11.10 - Evaluate the indefinite integral as an infinite...Ch. 11.10 - Evaluate the indefinite integral as an infinite...Ch. 11.10 - Evaluate the indefinite integral as an infinite...Ch. 11.10 - Use series to approximate the definite integral to...Ch. 11.10 - Prob. 64ECh. 11.10 - Use series to approximate the definite integral to...Ch. 11.10 - Use series to approximate the definite integral to...Ch. 11.10 - Use series to evaluate the limit. 61....Ch. 11.10 - Use series to evaluate the limit. 62....Ch. 11.10 - Use series to evaluate the limit. 63....Ch. 11.10 - Prob. 70ECh. 11.10 - Prob. 71ECh. 11.10 - Prob. 72ECh. 11.10 - Prob. 73ECh. 11.10 - Prob. 74ECh. 11.10 - Prob. 75ECh. 11.10 - Prob. 76ECh. 11.10 - Prob. 77ECh. 11.10 - Use multiplication or division of power series to...Ch. 11.10 - Prob. 79ECh. 11.10 - Find the function represented by the given power...Ch. 11.10 - Prob. 81ECh. 11.10 - Prob. 82ECh. 11.10 - Prob. 83ECh. 11.10 - Find the sum of the series. 74. n=0(1)n2n62n(2n)!Ch. 11.10 - Find the sum of the series. 75. n=1(1)n13nn5nCh. 11.10 - Find the sum of the series. 76. n=03n5nn!Ch. 11.10 - Find the sum of the series. 77....Ch. 11.10 - Find the sum of the series. 78....Ch. 11.10 - Find the sum of the series. 79. 3+92!+273!+814!+Ch. 11.10 - Find the sum of the series. 80. 1121323+15251727+Ch. 11.10 - Show that if p is an nth-degree polynomial, then...Ch. 11.10 - Prob. 92ECh. 11.10 - Use the Maclaurin series for f(x)=xsinx2 to find...Ch. 11.10 - If f(x) = ( 1 + x3)30, what is f(58)(0)?Ch. 11.10 - Prob. 95ECh. 11.10 - (a) Show that the function defined by...Ch. 11.10 - Use the followi ng steps to prove ( 17). (a) Let...Ch. 11.10 - Prob. 98ECh. 11.10 - Prob. 1DPCh. 11.11 - (a) Find the Taylor polynomials up to degree 5 for...Ch. 11.11 - (a) Find the Taylor polynomials up to degree 3 for...Ch. 11.11 - Find the Taylor polynomial T3(x) for the function...Ch. 11.11 - Find the Taylor polynomial T3(x) for the function...Ch. 11.11 - Find the Taylor polynomial T3(x) for the function...Ch. 11.11 - Find the Taylor polynomial T3(x) for the function...Ch. 11.11 - Find the Taylor polynomial T3(x) for the function...Ch. 11.11 - Prob. 8ECh. 11.11 - Find the Taylor polynomial T3(x) for the function...Ch. 11.11 - Find the Taylor polynomial T3(x) for the function...Ch. 11.11 - Prob. 11ECh. 11.11 - Prob. 12ECh. 11.11 - (a) Approximate f by a Taylor polynomial with...Ch. 11.11 - (a) Approximate f by a Taylor polynomial with...Ch. 11.11 - (a) Approximate f by a Taylor polynomial with...Ch. 11.11 - (a) Approximate f by a Taylor polynomial with...Ch. 11.11 - (a) Approximate f by a Taylor polynomial with...Ch. 11.11 - Prob. 18ECh. 11.11 - (a) Approximate f by a Taylor polynomial with...Ch. 11.11 - (a) Approximate f by a Taylor polynomial with...Ch. 11.11 - (a) Approximate f by a Taylor polynomial with...Ch. 11.11 - Prob. 22ECh. 11.11 - Use the information from Exercise 5 to estimate...Ch. 11.11 - Use the information from Exercise 16 to estimate...Ch. 11.11 - Use Taylors Inequality to determine the number of...Ch. 11.11 - How many terms of the Maclaurin series for ln(1 +...Ch. 11.11 - Use the Alternating Series Estimation Theorem or...Ch. 11.11 - Use the Alternating Series Estimation Theorem or...Ch. 11.11 - Use the Alternating Series Estimation Theorem or...Ch. 11.11 - Suppose you know that f(n)(4)=(1)nn!3n(n+1) and...Ch. 11.11 - A car is moving with speed 20 m/s and acceleration...Ch. 11.11 - The resistivity of a conducting wire is the...Ch. 11.11 - An electric dipole consists of two electric...Ch. 11.11 - (a) Derive Equation 3 for Gaussian optics from...Ch. 11.11 - Prob. 35ECh. 11.11 - A uniformly charged disk has radius R and surface...Ch. 11.11 - If a surveyor measures differences in elevation...Ch. 11.11 - The period of a pendulum with length L that makes...Ch. 11.11 - Prob. 39ECh. 11 - (a) What is a convergent sequence? (b) What is a...Ch. 11 - Prob. 2CCCh. 11 - Prob. 3CCCh. 11 - Suppose an=3 and sn, is the nth partial sum of the...Ch. 11 - Prob. 5CCCh. 11 - Prob. 6CCCh. 11 - Prob. 7CCCh. 11 - Prob. 8CCCh. 11 - Prob. 9CCCh. 11 - Prob. 10CCCh. 11 - Prob. 11CCCh. 11 - Write the binomial series expansion of ( 1 + x)k....Ch. 11 - Determine whether the statement is true or false....Ch. 11 - Determine whether the statement is true or false....Ch. 11 - Prob. 3TFQCh. 11 - Determine whether the statement is true or false....Ch. 11 - Determine whether the statement is true or false....Ch. 11 - Determine whether the statement is true or false....Ch. 11 - Determine whether the statement is true or false....Ch. 11 - Prob. 8TFQCh. 11 - Prob. 9TFQCh. 11 - Prob. 10TFQCh. 11 - Prob. 11TFQCh. 11 - Prob. 12TFQCh. 11 - Prob. 13TFQCh. 11 - Determine whether the statement is true or false....Ch. 11 - Determine whether the statement is true or false....Ch. 11 - Prob. 16TFQCh. 11 - Prob. 17TFQCh. 11 - Prob. 18TFQCh. 11 - Prob. 19TFQCh. 11 - Prob. 20TFQCh. 11 - Prob. 21TFQCh. 11 - Determine whether the statement is true or false....Ch. 11 - Determine whether the sequence is convergent or...Ch. 11 - Determine whether the sequence is convergent or...Ch. 11 - Determine whether the sequence is convergent or...Ch. 11 - Determine whether the sequence is convergent or...Ch. 11 - Prob. 5ECh. 11 - Prob. 6ECh. 11 - Determine whether the sequence is convergent or...Ch. 11 - Determine whether the sequence is convergent or...Ch. 11 - Prob. 9ECh. 11 - Prob. 10ECh. 11 - Determine whether the series is convergent or...Ch. 11 - Determine whether the series is convergent or...Ch. 11 - Determine whether the series is convergent or...Ch. 11 - Determine whether the series is convergent or...Ch. 11 - Determine whether the series is convergent or...Ch. 11 - Determine whether the series is convergent or...Ch. 11 - Prob. 17ECh. 11 - Prob. 18ECh. 11 - Prob. 19ECh. 11 - Prob. 20ECh. 11 - Prob. 21ECh. 11 - Determine whether the series is convergent or...Ch. 11 - Prob. 23ECh. 11 - Determine whether the series is conditionally...Ch. 11 - Determine whether the series is conditionally...Ch. 11 - Determine whether the series is conditionally...Ch. 11 - Find the sum of the series. 27. n=1(3)n123nCh. 11 - Find the sum of the series. 28. n=11n(n+3)Ch. 11 - Prob. 29ECh. 11 - Prob. 30ECh. 11 - Prob. 31ECh. 11 - Prob. 32ECh. 11 - Show that cosh x1+12x2 for all x.Ch. 11 - Prob. 34ECh. 11 - Prob. 35ECh. 11 - Prob. 36ECh. 11 - Prob. 37ECh. 11 - Prob. 38ECh. 11 - Prob. 39ECh. 11 - Find the radius of convergence and interval of...Ch. 11 - Find the radius of convergence and interval of...Ch. 11 - Prob. 42ECh. 11 - Prob. 43ECh. 11 - Prob. 44ECh. 11 - Prob. 45ECh. 11 - Prob. 46ECh. 11 - Find the Maclaurin series for f and the associated...Ch. 11 - Find the Maclaurin series for f and the associated...Ch. 11 - Find the Maclaurin series for f and the associated...Ch. 11 - Prob. 50ECh. 11 - Prob. 51ECh. 11 - Find the Maclaurin series for f and the associated...Ch. 11 - Prob. 53ECh. 11 - Prob. 54ECh. 11 - Prob. 55ECh. 11 - Prob. 56ECh. 11 - Prob. 57ECh. 11 - Prob. 58ECh. 11 - Prob. 59ECh. 11 - Prob. 60ECh. 11 - Prob. 61ECh. 11 - Prob. 62ECh. 11 - Prob. 1PCh. 11 - Let {Pn} be a sequence of points determined as in...Ch. 11 - Prob. 3PCh. 11 - Prob. 4PCh. 11 - Prob. 5PCh. 11 - Prob. 6PCh. 11 - Prob. 7PCh. 11 - Prob. 8PCh. 11 - Find the interval of convergence of n=1n3xn and...Ch. 11 - Suppose you have a large supply of books, all the...Ch. 11 - Find the sum of the series n=2ln(11n2).Ch. 11 - Prob. 12PCh. 11 - Prob. 13PCh. 11 - Prob. 14PCh. 11 - Prob. 15PCh. 11 - Prob. 16PCh. 11 - Prob. 17PCh. 11 - Prob. 18PCh. 11 - Prob. 19PCh. 11 - Prob. 20PCh. 11 - Prob. 21PCh. 11 - (a) Show that the Maclaurin series of the function...Ch. 11 - Let...Ch. 11 - Prob. 24P
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