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Q. 16

Expert-verified

Calculus

Found in: Page 624

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Use the divergence test to analyze the given series. Each answer should either be the series diverges or the divergence test fails, along with the reason for your answer.
$\sum_{k = 1}^{\infty} \frac{1}{k}$

The divergence test failed as $\lim_{k \to \infty} \frac{1}{k} = 0$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

Consider the given series,

$\sum_{k = 1}^{\infty} \frac{1}{k}$

Step 2. Analyze the series.

The divergence test states that if the sequence $\{a_{k}\}$ does not converge to zero, then the series ${\sum a_{k}}_{k = 1}^{\infty}$ diverges.

The value of the sequence $\{a_{k}\} = \{\frac{1}{k}\}$ is given below,

$\lim_{k \to \infty} a_{k} = \lim_{k \to \infty} \frac{1}{k} = 0$

Thus, the divergence test failed as $\lim_{k \to \infty} \frac{1}{k} = 0$ .

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