bounds for e

If $n$ and $m$ are positive integers and $n>m$ , we havethe following inequality:

\\left(1+{1\\over n}\\right)^{n}<{n\\over n+1}\\left(1+{1\\over m}\\right)^{m+1}

Taking the limit as $n\\to\\infty$ , we obtain an upper bound for $e$ .Combining this with the fact that the $(1+1/n)^{n}$ is an increasingsequence, we have the following bounds for $e$ :

\\left(1+{1\\over m}\\right)^{m}<e<\\left(1+{1\\over m}\\right)^{m+1}

This can be used to show that $e$ is not an integer – if we take $m=5$ , we obtain $2.48832<e<2.985984$ , for instance.