Exercises in Elementary Mathematics

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We solve a lot of exercises (from easy to difficult) using the computer algebra system Maple.

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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

Exercise 1. Let $\alpha\in[0,1]$ and define \[ a_1:=\frac{\alpha}{2},\quad a_{n+1}:=\frac{a_n^2+\alpha}{2}. \] Investigate the conver...
Stolz's Theorem

Theorem. Assume that $(x_n)_{n=1}^{\infty}$ and $(y_n)_{n=1}^{\infty}$ are two sequences of real numbers such that $y_n$ is strictly in...
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

Theorem. If $\alpha\geq 1$ then \[ (1+x)^{\alpha}\geq 1+\alpha x, \quad x\geq -1. \]
Vector triple product law

Theorem 1. If $\mathbf{a},\mathbf{b},\mathbf{c}\in\mathbf{R}^3$ then $\mathbf{a}\times(\mathbf{b}\times\mathbf{c})$ can be written in ...
The $\displaystyle{\left( 1+\frac{1}{x}\right)^x}$ function

$\DeclareMathOperator{\sgn}{sgn}$ Define the function \[ f(x):=\left( 1+\frac{1}{x}\right)^x \quad (x>0). \] We investigate some propert...
Linearly independent system

Definition. An $n\times n$ complex matrix $M$ is strictly diagonally dominant if \[ |m_{ii}|>\sum_{j\neq i}|m_{ij}|\\ (i=1,...,n). \]...
Calculation of $\displaystyle{\lim_{x_0}}f(x)^{g(x)}$

Theorem. Let $x_0$ be a finite real number, $\infty$ or $-\infty$. Assume that (a) $\displaystyle{\lim_{x_0}}\,f(x)=1$; (b) $\...
The problem of ladder

Problem. Find the longest ladder we can turn right in the corridor.
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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