# bounds for e

If $n$ and $m$ are positive integers and $n>m$, we have the 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 increasing sequence, we have the following bounds for $e$:

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

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

Title bounds for e BoundsForE 2013-03-22 15:48:48 2013-03-22 15:48:48 rspuzio (6075) rspuzio (6075) 7 rspuzio (6075) Theorem msc 33B99