Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Use either the divergence test or the integral test to determine whether the series in Exercises 32–43 converge or diverge. Explain why the series meets the hypotheses of the test you select.
36. $\sum_{k = 1}^{\infty} \frac{k}{k^{2} + 3}$

The series is divergent.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information

We have been given the series $\sum_{k = 1}^{\infty} \frac{k}{k^{2} + 3}$

We have to determine whether the series converge or diverge.

Step 2. Determine whether the series converge or diverge.

Consider function $f (x) = \frac{x}{3 + x^{2}}$

The function is continuous, decreasing, with positive terms.

All the conditions of integral test are fulfilled.

So, integral test is applicable.

Consider the integral localid="1649088344379" $\int_{x = 1}^{\infty} f (x) d x = \int_{x = 1}^{\infty} \frac{x}{3 + x^{2}} d x$

localid="1649088398964" $\int_{x = 1}^{\infty} f (x) d x = \lim_{k \to \infty} \int_{x = 1}^{k} \frac{x}{3 + x^{2}} d x = \frac{1}{2} \lim_{k \to \infty} \int_{u = 4}^{k^{2} + 3} \frac{d u}{u} (P u t 3 + x^{2} = u \Rightarrow 2 x d x = d u) = \frac{1}{2} \lim_{k \to \infty} {[\ln |u|]}_{4}^{k^{2} + 3} = \frac{1}{2} \lim_{k \to \infty} [\ln |k^{2} + 3| - \ln |4|] = \infty$

The integral diverges.

So, the series is divergent.

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Calculus

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Short Answer

Use either the divergence test or the integral test to determine whether the series in Exercises 32–43 converge or diverge. Explain why the series meets the hypotheses of the test you select.
36. $\sum_{k = 1}^{\infty} \frac{k}{k^{2} + 3}$

Step by Step Solution

Step 1. Given information

Step 2. Determine whether the series converge or diverge.

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Step by Step Solution

Step 1. Given information

Step 2. Determine whether the series converge or diverge.

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Use either the divergence test or the integral test to determine whether the series in Exercises 32–43 converge or diverge. Explain why the series meets the hypotheses of the test you select.
36. $\sum_{k = 1}^{\infty} \frac{k}{k^{2} + 3}$