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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 10.2.64
Chapter 10, Problem 10.2.64

Which series in Exercises 53–76 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.
∑ (from n = 1 to ∞) (1 − 1/n)ⁿ

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First, identify the general term of the series: \(a_n = \left(1 - \frac{1}{n}\right)^n\).
Next, analyze the behavior of the term \(a_n\) as \(n\) approaches infinity by finding the limit \(\lim_{n \to \infty} \left(1 - \frac{1}{n}\right)^n\).
Recall the known limit \(\lim_{n \to \infty} \left(1 - \frac{1}{n}\right)^n = e^{-1}\), which is a nonzero constant.
Since the terms \(a_n\) do not approach zero, apply the Divergence Test (also called the nth-term test for divergence), which states that if \(\lim_{n \to \infty} a_n \neq 0\), then the series \(\sum a_n\) diverges.
Conclude that the series \(\sum_{n=1}^\infty \left(1 - \frac{1}{n}\right)^n\) diverges because its terms do not tend to zero, so it does not converge and therefore does not have a finite sum.

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Key Concepts

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

Convergence and Divergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit; otherwise, it diverges. Determining convergence involves testing whether the terms decrease sufficiently fast and if the sum stabilizes as more terms are added.
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Convergence of an Infinite Series

Limit of the General Term

A necessary condition for series convergence is that the general term approaches zero as n approaches infinity. If the limit of the term (1 - 1/n)^n is not zero, the series must diverge by the Test for Divergence.
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One-Sided Limits

Behavior of the Term (1 - 1/n)^n

The expression (1 - 1/n)^n approaches 1/e as n becomes large. Understanding this limit helps determine the behavior of the series terms and whether the series can converge or diverge.
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Divergence Test (nth Term Test)
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