Joy of Mathematics

Convergence of series

October 28, 2020 / Leave a comment

For $v =\langle x_1, x_2, \dots, x_n \rangle \in \mathbb{R}^n$ we define $\left \| v \right \|_p = \left ( \sum \limits_{i=1}^{n} \left | x_i \right |^p \right )^{1/p}$ and $\left \| v \right \|_{\infty} = \max \limits_{1 \leq i \leq n} \left | x_i \right |$ . For which $p$ does the series

$\sum_{p=1}^{\infty} \left ( \left \| v \right \|_p - \left \| v \right \|_{\infty} \right)$

converge?

Solution

We may assume that $v$ is not the zero vector and $n>1$ otherwise the series is trivially convergent. Then, we show that the series is convergent if and only if there is exactly one component of maximal absolute value.

(a) If the above condition is satisfied then, without loss of generality, let $x_1$ be the component of maximal absolute value and let $\displaystyle t = \frac{\max \limits_{2 \leq i \leq n} \left | x_i \right |}{\left | x_i \right |} \in [0, 1]$ . Hence , as $p \rightarrow +\infty$

$\begin{align*} 0 &\leq \left \| v \right \|_p - \left \| v \right \|_{\infty} \\ &= \left \| v \right \|_{\infty} \left ( \left ( 1 + \left ( n-1 \right )t^p \right )^{1/p} - 1 \right ) \\ &=\left \| v \right \|_{\infty} \left ( \exp \left ( \frac{\ln \left ( 1+ (n-1) t^p \right )}{p} \right ) - 1\right ) \\ &\sim \left \| v \right \|_{\infty} \left ( n-1 \right ) \frac{t^p}{p} \end{align*}$

and the given series is convergent because $\displaystyle \sum_{p=1}^{\infty} \frac{t^p}{p} < + \infty$ .

(b) If the above condition is not satisfied, then there are at least $2$ components of maximal absolute value and therefore

$\begin{align*} \left \| v \right \|_p - \left \| v \right \|_{\infty} & \geq \left \| v \right \|_{\infty} \left ( 2^{1/p} -1 \right )\\ &= \left \| v \right \|_{\infty} \left ( \exp \left ( \frac{\ln 2}{p} \right ) -1 \right ) \\ &\sim \left \| v \right \|_{\infty} \frac{\ln 2}{p} \end{align*}$

and the given series is not convergent because $\displaystyle \sum_{p=1}^{\infty} \frac{1}{p} = +\infty$ .

Inequality

October 28, 2020 / Leave a comment

Let $\mathbb{R}^n$ be endowed with the usual product and the usual norm. If $v = (x_1, x_2, \dots, x_n) \in \mathbb{R}^n$ then we define $\sum v = x_1 +x_2 + \cdots + x_n$ . Prove that

$\left \| v \right \|^2 \left \| w \right \|^2 \geq \left ( v \cdot w \right )^2 + \frac{\left ( \left \| v \right \| \left | \sum w \right | - \left \| w \right \| \left |\sum v \right | \right )^2}{n}$

Huygen’s Inequality

October 28, 2020 / Leave a comment

Let $\alpha \in \left[0, \frac{\pi}{2} \right)$ . Prove that

$2 \sin \alpha + \tan \alpha \geq 3\alpha$

Solution

Let $f(\alpha) = 2 \sin \alpha + \tan \alpha$ . $f$ is twice differentiable with

$f''(\alpha) = -2 \sin \alpha +2 \tan \alpha + 2 \tan^3 \alpha \geq 0$

Hence $f$ is convex. The tangent at $(0, 0)$ has equation $y=3x$ . The result follows.

Sigma divisor sum

October 27, 2020 / Leave a comment

Let $\sigma$ denote the divisor function. Prove that

$\lim_{n \rightarrow +\infty} \frac{1}{n^2} \sum_{k=1}^{n} \sigma(k) = \frac{\pi^2}{12}$

Solution

We have successively:

$\begin{align*} \frac{1}{n^2} \sum_{k\leq n}\sigma(k) &= \frac {1}{n^2}\sum_{k \leq n}\sum_{q \mid k}q \\ &= \frac {1}{n^2}\sum_{q,d \mid qd \leq n}q \\ &= \frac {1}{n^2} \sum_{d\leq n} \sum_{q \leq n/d}q \\ &= \frac {1}{2n^2}\sum_{d \leq n} \left ( \frac {n^2}{d^2}+ \mathcal{O} \left ( \frac {n}{d}\right) \right ) \\ &= \frac{1}{2} \sum_{d \leq n} \frac{1}{d^2} + \mathcal{O} \left ( \frac{1}{n} \sum_{d \mid n} \frac{1}{d} \right ) \\ & = \frac{1}{2} \sum_{d \leq n} \frac{1}{d^2} + \mathcal{O} \left ( \frac{\ln n}{n} \right ) \\ & \rightarrow \frac{\pi^2}{12} \end{align*}$

Derivative at 0

October 11, 2020 / Leave a comment

Let

$f(x) = \frac{\sqrt{1+2x} \cdot \sqrt[4]{1+4x} \cdot \sqrt[6]{1+6x} \cdot \cdots \cdot \sqrt[100]{1+100x}}{\sqrt[3]{1+3x} \cdot \sqrt[5]{1+5x} \cdot \sqrt[7]{1+7x} \cdot \cdots \cdot \sqrt[101]{1+101x}}$