Series convergence of $\sum_1^\infty\frac{(n!)^2+(2n)^n}{n^{2n}}$

Question

$$\sum_1^\infty\frac{(n!)^2+(2n)^n}{n^{2n}}$$ Is there a way to test convergence of this series wihout splitting it ? (in case splitting is correct)

I deleted my answer because I do not answer the question about splitting. I apologize. Cheers :-) — Claude Leibovici, Dec 06 '14 at 14:13
Find out the order of the numerator: is $(n!)^2\ge (2n)^n$ in general? Then you can compare the greatest of the two with the denominator. — abiessu, Dec 06 '14 at 14:13

score 1 · Accepted Answer · answered Dec 06 '14 at 14:22

1

Show, using Stirling's approximation, that for $n\ge10$, we have $0\le(2n)^n\le(n!)^2$, and then use squeezing.

answered Dec 06 '14 at 14:22

Lucian

1 Answers1