convergence and value of Σ_(n=1) ^∞ (n^n /((n!)^2 ))


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Question Number 140982 by Mathspace last updated on 14/May/21

$c o n v e r g e n c e a n d v a l u e o f$ $\sum_{n = 1}^{\infty} \frac{n^{n}}{{(n!)}^{2}}$
Commented by mohammad17 last updated on 14/May/21

$U_{n} = \frac{n^{n}}{{(n!)}^{2}}, U_{n + 1} = \frac{{(n + 1)}^{n + 1}}{{((n + 1)!)}^{2}}$ ${l i m}_{n \to \infty} \frac{U_{n + 1}}{U_{n}} = {l i m}_{n \to \infty} \frac{{(n + 1)}^{n} (n + 1)}{{(n + 1)}^{2} {(n!)}^{2}} \times \frac{{(n!)}^{2}}{n^{n}}$ $= {l i m}_{n \to \infty} {(\frac{n + 1}{n})}^{n} \times {l i m}_{n \to \infty} \frac{1}{(n + 1)} = e^{n} \times 0 = 0$ $S o, t h e s e r i e s i s c o n v e r g e b y r a t i o t e s t$
Commented by mathmax by abdo last updated on 14/May/21

$why \lim_{n \to \infty} e^{n} . 0 = 0 ?$
Commented by mnjuly1970 last updated on 14/May/21

$e . 0 = 0$
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