"Use the Root Test to determine the convergence..." solved and explained

Answered question

2022-02-18

Use the Root Test to determine the convergence or divergence of the series.

\sum_{n = 1}^{\infty} (\frac{n}{3 n + 1})^{n}

Vasquez

Expert2022-03-12Added 669 answers

Given series $\sum_{n = 1}^{\infty} (\frac{n}{3 n + 1})^{n}$

There really isn’t much to these problems other than computing the limit and then using the root test. Here is the limit for this problem.

$L = lim_{n \to \infty} {| (\frac{n}{3 n + 1})^{n} |}^{\frac{1}{n}}$

$= lim_{n \to \infty} \frac{n}{3 n + 1}$

$= \frac{\infty}{\infty}$

$= 1$

So, by the Root Test this series is converges.

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