![\[ \boxed{x^\epsilon<1+\epsilon(x-1)}\]](https://www.adamponting.com/wp-content/ql-cache/quicklatex.com-91f0c1881b5d0ec16e9bcd412aa9dea9_l3.png "Rendered by QuickLaTeX.com")
where 
and 
.
When 
Derivation from the AM-GM inequality:
Using a set of 
positive numbers, 
of which have the value 
, and the 
others have the value 1:
The arithmetic mean is 
, so
![\[\sqrt[m]{a^n}<1+\frac{n}{m}(a-1)\]](https://www.adamponting.com/wp-content/ql-cache/quicklatex.com-8eb661e1d0889e4ffeba9eaaaff80f0d_l3.png "Rendered by QuickLaTeX.com")
Or, writing 
in place of 
,
![\[a^\epsilon<1+\epsilon(a-1)\]](https://www.adamponting.com/wp-content/ql-cache/quicklatex.com-81ef11bd29b0bb492611b2282c0e0a51_l3.png "Rendered by QuickLaTeX.com")
where and .
When
Derivation from the AM-GM inequality:
Using a set of positive numbers, of which have the value , and the others have the value 1:
The arithmetic mean is , so
Or, writing in place of ,