Exercises in Elementary Mathematics

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Snell's Law

A nice application of the calculation of extremal value.

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Labels: extreme values , functions , physics

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A remarkable sequence

Theorem 1. Let $x\in\mathbf{R}$, $x\gt 0$ is given. Then \[ \left( 1+\frac{x}{n}\right)^n\lt\left( 1+\frac{x}{n+1}\right)^{n+1} \] ...

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Recursive sequences

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Stolz's Theorem

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A remarkable sequence

Theorem 1. Let $x\in\mathbf{R}$, $x\gt 0$ is given. Then \[ \left( 1+\frac{x}{n}\right)^n\lt\left( 1+\frac{x}{n+1}\right)^{n+1} \] ...
Bernoulli's inequality

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Linearly independent system

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The problem of ladder

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Cesàro mean

Theorem. If the sequence $\{a_n\}$ is increasing, then their arithmetic mean \[ \sigma_n:=\frac{a_1+\ldots+a_n}{n} \] is also increas...

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