convergence-and-value-of-n-1-n-n-n-2-

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