Question 8.5.8: Using the Root Test Use the Root Test to determine the conve...
Using the Root Test Use the Root Test
to determine the convergence or divergence of the series \sum_{k=1}^{\infty}\left(\frac{2 k+4}{5 k-1}\right)^k.
In this case, we consider
\lim _{k \rightarrow \infty} \sqrt[k]{\left|a_k\right|}=\lim _{k \rightarrow \infty} \sqrt[k]{\left|\frac{2 k+4}{5 k-1}\right|^k}=\lim _{k \rightarrow \infty} \frac{2 k+4}{5 k-1}=\frac{2}{5}<1 .By the Root Test, the series is absolutely convergent.
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