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}

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
asked 2021-03-05
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.
\(\sum_{n=1}^\infty\frac{(2n)!}{n^5}\)

Answers (1)

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.
0

Relevant Questions

asked 2021-03-18
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.
\(\sum_{n=1}^\infty\frac{n^2}{(n+1)(n^2+2)}\)
asked 2020-12-28
Use the Direct Comparison Test or the Limit Comparison Test to determine the convergence or divergence of the series.
\(\sum_{n=1}^\infty\frac{1}{\sqrt{n^3+2n}}\)
asked 2021-03-08
Use the Limit Comparison Test to determine the convergence or divergence of the series.
\(\sum_{n=1}^\infty\frac{2n^2-1}{3n^5+2n+1}\)
asked 2021-01-13
Use the Direct Comparison Test to determine the convergence or divergence of the series.
\(\sum_{n=1}^\infty\frac{\sin^2n}{n^3}\)
asked 2020-12-16
Use the Limit Comparison Test to determine the convergence or divergence of the series.
\(\sum_{n=1}^\infty\frac{5}{4^n+1}\)
asked 2021-01-10
Use the Limit Comparison Test to prove convergence or divergence of the infinite series.
\(\displaystyle{\sum_{{{n}={1}}}^{\infty}}{\frac{{{e}^{{n}}+{n}}}{{{e}^{{{2}{n}}}-{n}^{{2}}}}}\)
asked 2020-10-25
Use the Alternating Series Test, if applicable, to determine the convergence or divergence of the series.
\(\sum_{n=2}^\infty\frac{(-1)^nn}{n^2-3}\)
asked 2021-03-02
Use the Alternating Series Test, if applicable, to determine the convergence or divergence of the series.
\(\sum_{n=1}^\infty\frac{(-1)^n}{n^5}\)
asked 2020-12-14
Use the limit Comparison Test to detemine the convergence or divergence of the series.
\(\sum_{n=1}^\infty\frac{5}{n+\sqrt{n^2+4}}\)
asked 2020-11-24
Use the limit Comparison Test to detemine the convergence or divergence of the series.
\(\displaystyle{\sum_{{{n}={1}}}^{\infty}}{\frac{{{5}}}{{{n}+\sqrt{{{n}^{{2}}+{4}}}}}}\)
...