Use ana appropriate test to determine whether the series converges. sum_{k=1}^infty(frac{k!}{20^kk^k})

Question
Series
asked 2020-12-02
Use ana appropriate test to determine whether the series converges.
\(\displaystyle{\sum_{{{k}={1}}}^{\infty}}{\left({\frac{{{k}!}}{{{20}^{{k}}{k}^{{k}}}}}\right)}\)

Answers (1)

2020-12-03
Given, the series is
\(\displaystyle{\sum_{{{k}={1}}}^{\infty}}{\left({\frac{{{k}!}}{{{20}^{{k}}{k}^{{k}}}}}\right)}\)
We have to check whether the series is convergent or divergent.
Use Ratio test,
If \(\displaystyle\sum{u}_{{k}}\) is a series of positive terms such that
\(\displaystyle\lim_{{{k}\to\infty}}{\frac{{{u}_{{k}}}}{{{u}_{{{k}+{1}}}}}}={l}\), then
\(\displaystyle\sum{u}_{{k}}\) is convergent if \(\displaystyle{l}{>}{1}\)
\(\displaystyle\sum{u}_{{k}}\) is divergent if\(\displaystyle{l}{<}{1}\)</span>
Test fail when \(\displaystyle{l}={1}\)
\(\displaystyle{u}_{{k}}={\frac{{{k}!}}{{{20}^{{k}}{k}^{{k}}}}}\) and \(\displaystyle{u}_{{{k}+{1}}}={\frac{{{\left({k}+{1}\right)}!}}{{{20}^{{{k}+{1}}}{\left({k}+{1}\right)}^{{{k}+{1}}}}}}\)
\(\displaystyle\lim_{{{k}\to\infty}}{\frac{{{u}_{{k}}}}{{{u}_{{{k}+{1}}}}}}=\lim_{{{k}\to\infty}}{\frac{{{\frac{{{k}!}}{{{20}^{{k}}{k}^{{k}}}}}}}{{{\frac{{{\left({k}+{1}\right)}!}}{{{20}^{{{k}+{1}}}{\left({k}+{1}\right)}^{{{k}+{1}}}}}}}}}\)
\(\displaystyle=\lim_{{{k}\to\infty}}{\frac{{{k}!{20}^{{{k}+{1}}}{\left({k}+{1}\right)}^{{{k}+{1}}}}}{{{20}^{{k}}{k}^{{k}}{\left({k}+{1}\right)}!}}}\)
\(\displaystyle=\lim_{{{k}\to\infty}}{\frac{{{k}!{20}{\left({k}+{1}\right)}^{{{k}}}{\left({k}+{1}\right)}}}{{{k}^{{k}}{\left({k}+{1}\right)}{k}!}}}\)
\(\displaystyle=\lim_{{{k}\to\infty}}{\frac{{{20}{\left({k}+{1}\right)}^{{k}}}}{{{k}^{{k}}}}}\)
\(\displaystyle=\lim_{{{k}\to\infty}}{20}{\left({1}+{\frac{{{1}}}{{{k}}}}\right)}^{{k}}\)
\(\displaystyle={20}{e}{>}{1}\)
Hence the series is convergent by Ratio test.
0

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