Chapter 8: Infinite Sequences and Series

Section 8.3: Convergence Tests

Example 8.3.7

$$ Determine if the series $\sum _{n\=1}^{\infty }\ln \left(\frac{n}{5 n\+2}\right)$ diverges, converges absolutely, or converges conditionally. If it converges conditionally, determine if it also converge absolutely. $$ Solution $$ That $\frac{n}{5 n\+2}\to \frac{1}{5}$ as $n\to \infty$ suggests considering the nth-term test. Indeed,       $\underset{n\to \infty }{lim}{a}_n\=\underset{n\to \infty }{lim}\ln \left(\frac{n}{5 n\+2}\right)\=-\ln \left(5\right)\ne 0$    so that the series necessarily diverges because the $n$th term does not go to zero.  $$

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