(1+αn)n is monotone for large n


Let α be a real number. The sequence (1+αn)n is monotone increasing for all n>|α|.


Let n>|α|. We want to prove the following inequalityMathworldPlanetmath:


Since both sides are positive, this follows by taking the (n+1)-th root and using the arithmetic-geometric-harmonic means inequality:

(1+αn)nn+1=1(1+αn)(1+αn)n+1n+1 elements1+n(1+αn)n+1=1+αn+1

Title (1+αn)n is monotoneMathworldPlanetmath for large n
Canonical name 1fracalphannIsMonotoneForLargeN
Date of creation 2013-03-22 17:53:55
Last modified on 2013-03-22 17:53:55
Owner uriw (288)
Last modified by uriw (288)
Numerical id 4
Author uriw (288)
Entry type Theorem
Classification msc 40-01
Classification msc 00-01