Use the Ratio Test to determine the convergence or divergence of the series. If the Ratio Test is inconclusive, determine the convergence or divergence of the series using other methods. sum_{n=1}^inftyfrac{(2n)!}{n^5}

Question
Series
Use the Ratio Test to determine the convergence or divergence of the series. If the Ratio Test is inconclusive, determine the convergence or divergence of the series using other methods.
$$\sum_{n=1}^\infty\frac{(2n)!}{n^5}$$

2021-03-06
The given series is $$\sum_{n=1}^\infty\frac{(2n)!}{n^5}$$
Compare the above series with its standard form $$\sum_{n=1}^\infty a_n$$ and obtain that $$a_n=\frac{(2n)!}{n^5}$$
Apply the ratio test as follows.
$$\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=\lim_{n\to\infty}\frac{(2n+2)!}{(n+1)^5}\times\frac{n^5}{(2n)!}$$
$$\lim_{n\to\infty}\frac{(2n+1)(2n+2)}{(1+\frac{1}{n})^5}$$
$$=\infty$$
$$>1$$
By ratio test,
the series $$\sum_{n=1}^\infty\frac{(2n)!}{n^5}$$ diverges.

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