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)

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.3: Geometric Sequences

Problem 81E

See similar textbooks