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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblem 10.2.39
Chapter 10, Problem 10.2.39

Telescoping Series
In Exercises 39–44, find a formula for the nth partial sum of the series and use it to determine if the series converges or diverges. If a series converges, find its sum.
∑ (from n = 1 to ∞) [ (1/n) − (1/(n + 1)) ]

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Identify the general term of the series: \( a_n = \frac{1}{n} - \frac{1}{n+1} \). This is a telescoping series because consecutive terms will cancel out when summed.
Write the nth partial sum \( S_n \) as the sum of the first n terms: \[ S_n = \sum_{k=1}^n \left( \frac{1}{k} - \frac{1}{k+1} \right) \].
Expand the partial sum to see the telescoping effect: \[ S_n = \left( \frac{1}{1} - \frac{1}{2} \right) + \left( \frac{1}{2} - \frac{1}{3} \right) + \cdots + \left( \frac{1}{n} - \frac{1}{n+1} \right) \]. Notice how most terms cancel out.
After cancellation, simplify the expression for \( S_n \) to find a formula involving only the first and last terms: \[ S_n = \frac{1}{1} - \frac{1}{n+1} \].
Determine the convergence of the series by taking the limit as \( n \to \infty \) of \( S_n \): \[ \lim_{n \to \infty} S_n = \lim_{n \to \infty} \left( 1 - \frac{1}{n+1} \right) \]. If this limit exists and is finite, the series converges to that value.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Telescoping Series

A telescoping series is a series whose partial sums simplify by canceling intermediate terms, leaving only a few terms from the beginning and end. This simplification makes it easier to find a formula for the nth partial sum and analyze convergence.
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Geometric Series

Partial Sums and Convergence

The nth partial sum is the sum of the first n terms of a series. Studying the limit of these partial sums as n approaches infinity helps determine if the series converges (approaches a finite value) or diverges (grows without bound).
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Integration Using Partial Fractions

Sum of a Convergent Series

If a series converges, its sum is the limit of its partial sums as n goes to infinity. For telescoping series, this sum can often be found by evaluating the simplified partial sum expression at the limits.
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Convergence of an Infinite Series
Related Practice
Textbook Question

Determining Convergence or Divergence

In Exercises 17–46, use any method to determine whether the series converges or diverges. Give reasons for your answer.

∑(from n=1 to ∞) [(-1)ⁿ n² e⁻ⁿ]

Textbook Question

Error Estimation

In Exercises 49–52, estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series.

1 / (1 + t) = ∑ (from n = 0 to ∞) [(-1)ⁿ tⁿ],0 < t < 1

Textbook Question

Limit Comparison Test

In Exercises 9–16, use the Limit Comparison Test to determine if each series converges or diverges.

∑ (from n=2 to ∞) 1 / ln n

(Hint: Limit Comparison with ∑ (from n=2 to ∞) (1/n))

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Textbook Question

Intervals of Convergence

In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?

∑ (from n = 0 to ∞) (2x)ⁿ

Textbook Question

Are there any values of x for which ∑ (from n=1 to ∞) (1 / nˣ) converges? Give reasons for your answer.

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Textbook Question

Determining Convergence or Divergence

In Exercises 1–14, determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.

∑ (from n = 2 to ∞) [(-1)ⁿ⁺¹ (1 / ln n)]

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