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

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

(Theorem)

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

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

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

Since both sides are positive, this follows by taking the

-th root and using the arithmetic-geometric-harmonic means inequality:

$\displaystyle \sqrt[n+1]{\left(1+\frac{\alpha}{n}\right)^n} = \underbrace{\sqrt... ...} } \leq \frac{1+n\left(1+\frac{\alpha}{n}\right)}{n+1} = 1+\frac{\alpha}{n+1}$

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Cross-references: arithmetic-geometric-harmonic means inequality, root, positive, sides, inequality, monotone increasing, sequence, real number

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

, born on 2008-03-12.
Object id is 10389, canonical name is 1fracalphannIsMonotoneForLargeN.
Accessed 280 times total.

Classification:

AMS MSC:	40-01 (Sequences, series, summability :: Instructional exposition )
	00-01 (General :: Instructional exposition )

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