Find-

Find-

Question Number 206845 by hardmath last updated on 27/Apr/24

Find : \frac{\infty!}{\infty^{\infty}} = ?

Commented by A5T last updated on 27/Apr/24

D o y o u w i s h t o f i n d t h i s : \underset{n \to \infty}{l i m} \frac{n!}{n^{n}} ?

Commented by hardmath last updated on 27/Apr/24

Yes Ser

Commented by A5T last updated on 27/Apr/24

0

Answered by Frix last updated on 27/Apr/24

\lim_{n \to \infty} \frac{n!}{n^{n}} \underset{Formula]}{\overset{[{Sterling}^{'} s}{=}} \lim_{n \to \infty} \frac{\frac{n^{n}}{e^{n}} \sqrt{2 π n}}{n^{n}} = \lim_{n \to \infty} \frac{\sqrt{2 π n}}{e^{n}} = 0

Commented by hardmath last updated on 28/Apr/24

thank you dear Ser

Commented by MM42 last updated on 30/Apr/24

\forall n > 2 \to 0 < \frac{n!}{n^{n}} < \frac{n!}{(n + 1)!} = \frac{1}{n + 1}

\Rightarrow {l i m}_{n \to \infty} \frac{n!}{n^{n}} = 0 ✓

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