${(1+\frac{\alpha}{n})}^{n}$ is monotone for large $n$
Lemma.
Let $\alpha $ be a real number. The sequence ${\mathrm{(}\mathrm{1}\mathrm{+}\frac{\alpha}{n}\mathrm{)}}^{n}$ is monotone increasing for all $n\mathrm{>}\mathrm{|}\alpha \mathrm{|}$.
Proof.
Let $n>|\alpha |$. We want to prove the following inequality^{}:
$${\left(1+\frac{\alpha}{n}\right)}^{n}\le {\left(1+\frac{\alpha}{n+1}\right)}^{n+1}$$ |
Since both sides are positive, this follows by taking the $(n+1)$-th root and using the arithmetic-geometric-harmonic means inequality:
$$\sqrt[n+1]{{\left(1+\frac{\alpha}{n}\right)}^{n}}=\underset{n+1\text{elements}}{\underset{\u23df}{\sqrt[n+1]{1\cdot \left(1+\frac{\alpha}{n}\right)\mathrm{\cdots}\left(1+\frac{\alpha}{n}\right)}}}\le \frac{1+n\left(1+\frac{\alpha}{n}\right)}{n+1}=1+\frac{\alpha}{n+1}$$ |
∎
Title | ${(1+\frac{\alpha}{n})}^{n}$ is monotone^{} for large $n$ |
---|---|
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 |