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Expert-verified Found in: Page 624 ### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861 # 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 $\underset{k\to \infty }{\mathrm{lim}}\frac{1}{k}=0$.

See the step by step solution

## 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 $\left\{{a}_{k}\right\}$ does not converge to zero, then the series $\underset{k=1}{\overset{\infty }{\sum {a}_{k}}}$ diverges.

The value of the sequence $\left\{{a}_{k}\right\}=\left\{\frac{1}{k}\right\}$ is given below,

$\underset{k\to \infty }{\mathrm{lim}}{a}_{k}=\underset{k\to \infty }{\mathrm{lim}}\frac{1}{k}\phantom{\rule{0ex}{0ex}}=0$

Thus, the divergence test failed as $\underset{k\to \infty }{\mathrm{lim}}\frac{1}{k}=0$. ### Want to see more solutions like these? 