Show that $ \overline \lim\limits_{n \rightarrow \infty}\left(\frac{1+a_{n+1}}{a_{n}}\right)^{n} \geq \mathrm{e} $

Asked Feb 19 '19 at 13:38

Active Feb 19 '19 at 21:51

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Show that $$ \overline \lim\limits_{n \rightarrow \infty}\left(\frac{1+a_{n+1}}{a_{n}}\right)^{n} \geq \mathrm{e} $$ For any sequence $\{a_n\}$ with positive terms, and that this estimate cannot be improved.

I found this problem in "Mathematical Analysis I" of Zorich, just Google Books and turn to page 149 of the preview. I believe this problem is somehow incorrectly stated (since I can't solve it :) ) , so I asked for sure.

Thanks.

edited Feb 19 '19 at 21:51

rtybase

16,907

asked Feb 19 '19 at 13:38

Li Taiji

Another one: https://math.stackexchange.com/q/747987/42969 – Both found with Approach0 – Martin R Feb 19 '19 at 14:01
Good.Thank you.@ Approach0 @ Martin R – Li Taiji Feb 19 '19 at 14:40
@Martin R I need to delete this question? – Li Taiji Feb 19 '19 at 14:47

Show that $ \overline \lim\limits_{n \rightarrow \infty}\left(\frac{1+a_{n+1}}{a_{n}}\right)^{n} \geq \mathrm{e} $

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