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}<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.

Title	bounds for e
Canonical name	BoundsForE
Date of creation	2013-03-22 15:48:48
Last modified on	2013-03-22 15:48:48
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	7
Author	rspuzio (6075)
Entry type	Theorem
Classification	msc 33B99