Does $\sum_1^\infty\frac{(n!)^2+(2n)^n}{n^{2n}}$ converge? (without Stirling's approximation)

Question

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.

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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.

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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

Answer 1

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.

Answer 2

Convergence of this series is quite easy. First note that $\sum \frac {(2n)^{n}} {n^{2n}}$ is convergent since it is dominated by $\frac {2^{n}} {3^{n}}$ if you omit the first two terms. Now use the fact that $n! < (1)(2)(n^{n-2})$ for $n \geq 3$. Can you finish?

Answer 3

We have that eventually, using this result

$$(n!)^2\ge \left(\frac{n^2}{e^2}\right)^n\ge(2n)^n$$

therefore

$$\frac{(n!)^2+(2n)^n}{n^{2n}}\le \frac{2(n!)^2}{n^{2n}}=2\frac{n!}{n^{n}}\frac{n!}{n^{n}} \le \frac2{n^2}$$

since

$$\frac{n!}{n^{n}}=\frac{1\cdot 2\cdots n}{n\cdot n\cdots n} \le \frac{1}{n}\cdot 1\cdot 1\cdots 1 = \frac{1}{n}$$