# Chapter 9 - Section 9.5 - Absolute Convergence; The Ratio and Root Tests - Exercises - Page 515: 4

Converges

#### Work Step by Step

Consider $a_n=\dfrac{2^{n+1}}{n3^{n-1}}$ Now, $l=\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n+1}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{2^{n+2}}{(n+1)3^{n}}}{\dfrac{2^{n+1}}{n3^{n-1}}}|$ Thus, we have $l=\lim\limits_{n \to \infty}|\dfrac{2n}{3(n+1)}|=\lim\limits_{n \to \infty}|\dfrac{2n}{3n+3}|=\dfrac{2}{3} \lt 1$ Hence, the series Converges absolutely by the ratio test.

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