$(1+\frac{\alpha}{n})^{n}$ is monotone for large $n$

Lemma.

Let $\alpha$ be a real number. The sequence $(1+\frac{\alpha}{n})^{n}$ is monotone increasing for all $n>|\alpha|$.

Proof.

Let $n>|\alpha|$. We want to prove the following inequality:

 $\left(1+\frac{\alpha}{n}\right)^{n}\leq\left(1+\frac{\alpha}{n+1}\right)^{n+1}$

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

 $\sqrt[n+1]{\left(1+\frac{\alpha}{n}\right)^{n}}=\underbrace{\sqrt[n+1]{1\cdot% \left(1+\frac{\alpha}{n}\right)\cdots\left(1+\frac{\alpha}{n}\right)}}_{% \textrm{n+1 elements}}\leq\frac{1+n\left(1+\frac{\alpha}{n}\right)}{n+1}=1+% \frac{\alpha}{n+1}$

Title $(1+\frac{\alpha}{n})^{n}$ is monotone for large $n$ 1fracalphannIsMonotoneForLargeN 2013-03-22 17:53:55 2013-03-22 17:53:55 uriw (288) uriw (288) 4 uriw (288) Theorem msc 40-01 msc 00-01