This question have been asked again a long ago here, but its only answer gives just a hint about Stirling's approximation.
I am trying to study the convergence $\sum_1^\infty\frac{(n!)^2+(2n)^n}{n^{2n}}$, but without Stirling's approximation.
I tried Cauchy's condensation test with no luck. Wolfram alpha suggests root test
The limit is:
$ \lim_{n\to\infty} \sqrt[n]{\frac{(n!)^2+(2n)^n}{n^{2n}}}$
Again Wolfram calulates the limit to be: $e^{-2}$
Therefore, somehow $ \lim_{n\to\infty}\sqrt[n]{\frac{(n!)^2+(2n)^n}{n^{2n}}} = \lim_{n\to\infty}
\left(1-\frac2{n}\right)^n$
but I have no idea how to prove that.
I don't want to reduce this question exclusively to the calculation of this limit, any answer that doesn't include Stirling's approximation is welcome.
Thanks
Best Answer
Note that$$\require{cancel}\frac{\frac{(n+1)!^2}{(n+1)^{2(n+1)}}}{\frac{n!^2}{n^{2n}}}=\cancel{(n+1)^2}\left(\frac{n}{n+1}\right)^{2n}\frac1{\cancel{(n+1)^2}}\to\frac1{e^2}$$and that$$\frac{\frac{(2(n+1))^{n+1}}{(n+1)^{2(n+1)}}}{\frac{(2n)^n}{n^{2n}}}=2(n+1)\left(\frac{n+1}n\right)^n\left(\frac n{n+1}\right)^{2n}\frac1{(n+1)^2}=\frac2{n+1}\left(\frac n{n+1}\right)^n\to0.$$So, both series$$\sum_{n=1}^\infty\frac{n!^2}{n^{2n}}\text{ and }\sum_{n=1}^\infty\frac{(2n)^n}{n^{2n}}$$converge and therefore so does their sum.