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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.AAE.30b
Chapter 10, Problem 10.AAE.30b

30. b. By differentiating the series in part (a) term by term, show that
Σ(from n=1 to ∞) n / (n + 1)! = 1.

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1
Recall the series from part (a), which is likely the sum of the reciprocals of factorials, such as \( \sum_{n=0}^\infty \frac{1}{n!} = e \). Identify the exact series given in part (a) to differentiate term by term.
Express the series \( \sum_{n=1}^\infty \frac{n}{(n+1)!} \) in a form that relates to the series from part (a). Notice that \( \frac{n}{(n+1)!} = \frac{n+1-1}{(n+1)!} = \frac{n+1}{(n+1)!} - \frac{1}{(n+1)!} = \frac{1}{n!} - \frac{1}{(n+1)!} \).
Rewrite the series using this decomposition: \( \sum_{n=1}^\infty \frac{n}{(n+1)!} = \sum_{n=1}^\infty \left( \frac{1}{n!} - \frac{1}{(n+1)!} \right) \).
Recognize that this is a telescoping series, where most terms cancel out when expanded. Write out the first few terms explicitly to see the cancellation pattern.
Sum the telescoping series by taking the limit as \( n \to \infty \) of the partial sums, and conclude that the sum converges to 1.

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Key Concepts

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

Term-by-Term Differentiation of Series

Term-by-term differentiation allows differentiating an infinite series by differentiating each term individually, provided the series converges uniformly. This technique is useful to find new series representations or to simplify expressions involving series.
Recommended video:
05:44
Divergence Test (nth Term Test)

Factorials and Series Manipulation

Factorials (n!) grow rapidly and often appear in series expansions like those of exponential functions. Understanding how to manipulate terms involving factorials, such as rewriting n/(n+1)! as n!/(n+1)!, is key to simplifying and evaluating series.
Recommended video:
5:22
Factorials

Convergence of Infinite Series

Determining whether an infinite series converges is essential before performing operations like differentiation term-by-term. Recognizing convergence ensures the validity of the manipulations and that the series sums to a finite value, such as the sum equaling 1 in this problem.
Recommended video:
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Convergence of an Infinite Series
Related Practice
Textbook Question

Theory and Examples

Suppose that a₁, a₂, a₃, …, aₙ are positive numbers satisfying the following conditions:

i) a₁ ≥ a₂ ≥ a₃ ≥ …;


ii) the series a₂ + a₄ + a₈ + a₁₆ + … diverges.

Show that the series


a₁/1 + a₂/2 + a₃/3 + …


diverges.

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Textbook Question

Power Series

In Exercises 47–56, (a) find the series’ radius and interval of convergence. Then identify the values of x for which the series converges (b) absolutely and (c) conditionally.


∑ (from n = 1 to ∞) (x + 4)ⁿ/(n3ⁿ)

Textbook Question

Determining Convergence or Divergence

Which of the series in Exercises 13–46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

∑ (from n=1 to ∞) eⁿ / (1 + e²ⁿ)

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Textbook Question

Power Series

In Exercises 47–56, (a) find the series’ radius and interval of convergence. Then identify the values of x for which the series converges (b) absolutely and (c) conditionally.

∑ (from n = 1 to ∞) (csch n)xⁿ

Textbook Question

Use series to evaluate the limits in Exercises 29–40.

37. lim (x → 0) ln(1 + x²) / (1 - cos(x))

Textbook Question

Applying the Integral Test

Use the Integral Test to determine if the series in Exercises 1–12 converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.

∑ (from n = 1 to ∞) 1 / n⁰·²

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