Limits

**Julius** · February 26th 2010, 07:08 AM

Find the limit:
$\lim\limits_{n\to\infty}(\frac{(1+\frac{1}{n})^n}{ e})^n$
the answer is $\frac{1}{\sqrt{e}}$ so i guess we should get something like $\lim\limits_{n\to\infty}(1-\frac{1}{2n})$ .. but dont know what to do with double n and exp. also:
$\lim\limits_{n\to\infty}(\prod\limits_{k=1}^{n}\fr ac{k}{k+n})^{e^{\frac{1999}{n}}-1}$ so thats $\lim\limits_{n\to\infty}(\frac{n!^2}{2n!})^{e^{\fr ac{1999}{n}}-1}$ but again, what to do with exp?

**Laurent** · February 26th 2010, 07:30 AM

Originally Posted by Julius

Find the limit:
$\lim\limits_{n\to\infty}(\frac{(1+\frac{1}{n})^n}{ e})^n$
the answer is $\frac{1}{\sqrt{e}}$ so i guess we should get something like $\lim\limits_{n\to\infty}(1-\frac{1}{2n})$ .. but dont know what to do with double n and exp.

This one is a particular case of the question in this thread.

lso:
$\lim\limits_{n\to\infty}(\prod\limits_{k=1}^{n}\fr ac{k}{k+n})^{e^{\frac{1999}{n}}-1}$ so thats $\lim\limits_{n\to\infty}(\frac{n!^2}{2n!})^{e^{\fr ac{1999}{n}}-1}$ but again, what to do with exp?

First of all, start by writing the sequence as $\exp((e^{\frac{1999}{n}}-1)\log\prod_{k=1}^n \frac{k}{k+n})$ . Then, use asymptotic expansions in the exponent. Note that $\log\prod_{k=1}\frac{k}{k+n}=\sum_{k=1}^n \log\frac{k/n}{k/n+1}$ , which evokes a Riemann sum; thus we can avoid Stirling's approximation.

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