Proving a particularly unconventional sequence is increasing

Asked Jan 21 '24 at 09:54

Active Jan 21 '24 at 09:54

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The task is to prove the following sequence $$\frac{e^n n!}{n^n}$$

is increasing by taking the ratio of successive elements, except when I take the ratio of the successive elements I get

$$e\bigl(\frac{n}{n+1}\bigr)^n$$

which I know is always greater than $1$ just by graphing it and I know it actually converges to $1$; yet when I try to prove it via induction I keep frustratingly hitting dead ends, I've been at this for a while to no avail.

Could anyone offer some guidance? Thank you.

asked Jan 21 '24 at 09:54

altayir1

$(1+1/n)^n < e$ is a well-known inequality, see for example https://math.stackexchange.com/q/1950375/42969. – Btw, what is “particularly unconventional” about your sequence? – Martin R Jan 21 '24 at 09:59
Also a duplicate of https://math.stackexchange.com/q/2278416/42969 – Martin R Jan 21 '24 at 10:03
1

The Stirling approximation should be helpful here. – Peter Jan 21 '24 at 10:05
I think I've got it, I just needed a fresh perspective. Thank you – altayir1 Jan 21 '24 at 10:08

Proving a particularly unconventional sequence is increasing

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