Question

True or false.Give an explanation for your answer. If the terms, $a_{n},$ of a series tend to zero as $n$ increases, then the series $\sum a_{n}$ converges.

Name: Problem 56, Chapter 9: Calculus Single Variable
Uploaded: 2021-05-02T19:44:37-08:00
Duration: 2 min 22 s
Channel: Priyanka Sadarangani
Description: True or false.Give an explanation for your answer. If the terms, an, of a series tend to zero as n increases, then the series ∑an converges.

   True or false.Give an explanation for your answer.
If the terms, $a_{n},$ of a series tend to zero as $n$ increases, then the series $\sum a_{n}$ converges.

Calculus Single Variable

Deborah… 7th Edition

Chapter 9, Problem 56

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Solved by Expert Priyanka Sadarangani

05/02/2021

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True or false.Give an explanation for your answer. If the terms, $a_{n},$ of a series tend to zero as $n$ increases, then the series $\sum a_{n}$ converges.

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Key Concepts

Insufficiency of Terms Approaching Zero for Convergence

Although the terms approaching zero is required, it is not a sufficient condition for the series to converge. A classic example is the harmonic series, where the terms approach zero, yet the series diverges. This illustrates that additional conditions or tests are necessary to confirm the convergence of a series.

Necessary Condition for Series Convergence

For an infinite series to converge, it is required that its terms approach zero as the number of terms increases. This is a necessary condition because if the terms do not tend toward zero, the series must diverge. However, this condition alone does not suffice to ensure convergence.

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