
(1) 
where
,
. This inequality can be proven by taking a Maclaurin Series
of ,

(2) 
Since the series terminates after a finite number of terms for Integral , the Bernoulli inequality
for is obtained by truncating after the firstorder term. When , slightly more finesse is needed. In this
case, let so that , and take

(3) 
Since each Power of multiplies by a number and since the Absolute Value of the Coefficient of
each subsequent term is smaller than the last, it follows that the sum of the third order and subsequent terms is a Positive
number. Therefore,

(4) 
or

(5) 
completing the proof of the Inequality over all ranges of parameters.
© 19969 Eric W. Weisstein
19990526