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
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![A telescoping series is a series in which each term \( a_n \) can be written as \( a_n = b_n - b_{n+1} \) for some sequence \( b_n \). 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 \( s_n \). Using \( a_n = b_n - b_{n+1} \), show that
\[
s_n = a_1 + a_2 + a_3 + \ldots + a_n = b_1 - b_{n+1}.
\]
**(b)** Using the limit of \( s_n \) as \( n \to \infty \), show that the series converges if \( \lim_{n \to \infty} b_{n+1} \) exists.
**(c)** Use partial fractions to show that
\[
\sum_{n=1}^{\infty} \frac{1}{n^2 + n} = \sum_{n=1}^{\infty} \left[ \frac{1}{n} - \frac{1}{n+1} \right]
\]
**(d)** Observe that the series in part (c) is telescoping (why?). Show that it converges using part (b).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa1d5cb46-a845-43db-921b-209c3c017b1f%2Fa8c29be4-99db-46d0-9adc-a26ba0dcf747%2Fjcj4yu4_processed.png&w=3840&q=75)
Transcribed Image Text:A telescoping series is a series in which each term \( a_n \) can be written as \( a_n = b_n - b_{n+1} \) for some sequence \( b_n \). 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 \( s_n \). Using \( a_n = b_n - b_{n+1} \), show that
\[
s_n = a_1 + a_2 + a_3 + \ldots + a_n = b_1 - b_{n+1}.
\]
**(b)** Using the limit of \( s_n \) as \( n \to \infty \), show that the series converges if \( \lim_{n \to \infty} b_{n+1} \) exists.
**(c)** Use partial fractions to show that
\[
\sum_{n=1}^{\infty} \frac{1}{n^2 + n} = \sum_{n=1}^{\infty} \left[ \frac{1}{n} - \frac{1}{n+1} \right]
\]
**(d)** Observe that the series in part (c) is telescoping (why?). Show that it converges using part (b).
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