Chapter 8: Infinite Sequences and Series

Section 8.2: Series

Example 8.2.7

$$ Test the series $\sum _{n\=1}^{\infty }\ln \left(\frac{n}{5n\+2}\right)$ for convergence. $$ Solution $$ Determine the limiting behavior of $a_n\=\ln \left(\frac{n}{5 n\+2}\right)$  •  Calculus palette: Limit template≻Apply to $\arctan \left(n\right)$ •  Context Panel: Evaluate and Display Inline $\underset{n\to \infty }{lim}\ln \left(\frac{n}{5 n\+2}\right)$ = $-\ln \left(5\right)$$$ $$ Because $a_n\=\ln \left(\frac{n}{5 n\+2}\right)\to -\ln \left(5\right)$ and not 0 as $n\to \infty$, the series cannot converge. Hence it must diverge.

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