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Tutorial Exercise
Use the Root Test to determine the convergence or divergence of the series.
$\sum_{n=1}^{\infty} \left(\frac{-4n}{3n+1}\right)^{2n}$
Step 1
Recall the Root Test, which states that if $\sum a_n$ is a series, and $\lim_{n \to \infty} \sqrt[n]{|a_n|} < 1$, then $\sum a_n$ converges
$\qquad$ diverges.
Step 2
For this series, $a_n = \left(\frac{-4n}{3n+1}\right)^{2n}$. Find $\lim_{n \to \infty} \sqrt[n]{|a_n|}.$
$\lim_{n \to \infty} \sqrt[n]{|a_n|} = \lim_{n \to \infty} \sqrt[n]{\left|\left(\frac{-4n}{3n+1}\right)^{2n}\right|} $
$= \lim_{n \to \infty} \left(\frac{4n}{3n+1}\right)^2$
$= \lim_{n \to \infty} \left(\frac{4}{3+\frac{1}{n}}\right)^2$
$= \left(\frac{4}{3}\right)^2$
$= \frac{16}{9}$
