Help with a limit calculation

[shahar1231]

Hi,
I'm having troubles calculating the next limit(it is a Rabbe's test for a series).


Thanks!

 

[Deleted member 4993]

Hi,
I'm having troubles calculating the next limit(it is a Rabbe's test for a series).
View attachment 21879

Thanks!

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[pka]

I suggest that we rearrange the limit:
\(\mathop {\lim }\limits_{n \to \infty } n\left( {1 - \frac{{{{\left( {1 + \frac{1}{{n + 1}}} \right)}^{{{(n + 1)}^2}}}}}{{e{{\left( {1 + \frac{1}{n}} \right)}^{{n^2}}}}}} \right) = \mathop {\lim }\limits_{n \to \infty } n\left( {1 + \frac{1}{e}{{\left( {1 + \frac{1}{n}} \right)}^{ - {n^2}}}{{\left( {1 + \frac{1}{{n + 1}}} \right)}^{{{(n + 1)}^2}}}} \right)\).
Now what is \(\mathop {\lim }\limits_{n \to \infty } \frac{1}{e}\left[ {{{\left( {1 + \frac{1}{n}} \right)}^{ - {n^2}}}} \right] = \frac{1}{e}\mathop {\lim }\limits_{n \to \infty } {\left[ {{{\left( {1 + \frac{1}{n}} \right)}^n}} \right]^{ - n}} = ?\) SEE HERE