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A telescoping series is a series in which each term an can be written as an = bn-bn+1 for some sequence bn. These patterns will more than often cause mass cancellation, making the problem solvable by hand. Some patterns are harder to find than others. Often, partial fractions are used to detect the pattern. (a) The benefit of telescoping series is that it allows us to easily add up the terms, leading to a simple formula for the partial sums sn. Using an = bn -bn+1 show that Sn = a₁ + a2 + a3 +...+ an = b₁ bn+1. n→∞ (b) Using the limit of sn as n → ∞, show that the series converges if lim bn+1 exists. (c) Use partial fractions to show that Σ n=1 1 n² + n = ∞ n=1 1 n n+ (d) Observe that the series in part (c) is telescoping (why?). Show that it converges using part (b)

A telescoping series is a series in which each term an can be written as an = bn-bn+1 for some sequence bn. These patterns will more than often cause mass cancellation, making the problem solvable by hand. Some patterns are harder to find than others. Often, partial fractions are used to detect the pattern. (a) The benefit of telescoping series is that it allows us to easily add up the terms, leading to a simple formula for the partial sums sn. Using an = bn -bn+1 show that Sn = a₁ + a2 + a3 +...+ an = b₁ bn+1. n→∞ (b) Using the limit of sn as n → ∞, show that the series converges if lim bn+1 exists. (c) Use partial fractions to show that Σ n=1 1 n² + n = ∞ n=1 1 n n+ (d) Observe that the series in part (c) is telescoping (why?). Show that it converges using part (b)

College Algebra
College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter8: Sequences And Series
Section: Chapter Questions
Problem 6CC
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A telescoping series is a series in which each term an can be written as an bn - bn+1 for some sequence bn. These patterns will more than often cause mass cancellation, making the problem solvable by hand. Some patterns are harder to find than others. Often, partial fractions are used to detect the pattern. - (a) The benefit of telescoping series is that it allows us to easily add up the terms, leading to a simple formula for the partial sums sn. Using an = bn bn+1 show that Sn = a₁ + a2 + a3 + ... + an = b₁ bn+1. (b) Using the limit of sn as n → ∞, show that the series converges if lim bn+1 exists. n→∞ (c) Use partial fractions to show that 1 1 Στις 24 n² + n n+ 1 n=1 n=1 = (d) Observe that the series in part (c) is telescoping (why?). Show that it converges using part (b).
Transcribed Image Text:A telescoping series is a series in which each term an can be written as an bn - bn+1 for some sequence bn. These patterns will more than often cause mass cancellation, making the problem solvable by hand. Some patterns are harder to find than others. Often, partial fractions are used to detect the pattern. - (a) The benefit of telescoping series is that it allows us to easily add up the terms, leading to a simple formula for the partial sums sn. Using an = bn bn+1 show that Sn = a₁ + a2 + a3 + ... + an = b₁ bn+1. (b) Using the limit of sn as n → ∞, show that the series converges if lim bn+1 exists. n→∞ (c) Use partial fractions to show that 1 1 Στις 24 n² + n n+ 1 n=1 n=1 = (d) Observe that the series in part (c) is telescoping (why?). Show that it converges using part (b).
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