A log integral

Prove that:

$$\int_{0}^{\pi/2}{x\log \left( 1-\cos x \right) \, {\rm d} x}=\frac{35}{16}\zeta \left( 3 \right)-\frac{\pi ^{2}}{8}\log 2-\pi\mathcal{G}$$

Solution

A trigonometric value

Evaluate

$$\mathcal{V}= \tan \left( 2 \arcsin \frac{1}{\sqrt{5}}\right)$$

Solution

On a known inequality

Let $x$ be a real number and let $n \in \mathbb{N}$. Prove the inequality:

$$\left | \sum_{k=1}^{n} \frac{\sin kx}{k} \right |\leq 2 \sqrt{\pi}$$

Solution

Expected value of a summation

Let $r_n$ be a random variable that returns one of the digits $2, 0, 1, 6$ each with equal probability for all positive integers $n$. Find the value of

$$\mathcal{V}=\mathbb{E} \left[ \sum_{n=1}^{\infty} \frac{r_n}{10^n} \right]$$

where $\mathbb{E}$ denotes the expected value of $x$.

Solution

Finite matrix group

Let $\mathcal{G}$ be a finite subgroup of ${\rm GL}_n (\mathbb{C})$ (this is the group of the $n \times n$ invertible matrices over $\mathbb{C}$). If $\sum \limits_{g \in \mathcal{G}} {\rm Tr}(g)=0$ then prove that $\sum \limits_{g \in \mathcal{G}} g =0$.

Solution

Integral with log and trigonometric

Prove that

$$\int_0^{\infty} \frac{\log t (1-\cos t)}{t^2}\, {\rm d}t = \frac{\pi}{2} (1-\gamma)$$

where $\gamma$ stands for the Euler - Mascheroni constant.

Solution

Trigonometric sum

Evaluate the value of

$$\mathcal{A}=\cos \frac{2\pi}{13} + \cos \frac{6 \pi}{13} + \cos \frac{8\pi}{13}$$

Solution

Series with trilogarithm

Let ${\rm Li}_3$ denote the trilogarithm function. Prove that:

$$\sum_{n=1}^{\infty} {\rm Li}_3 \left(e^{-2n \pi} \right)= \frac{7 \pi^3}{360} - \frac{\zeta(3)}{2}$$

(Seraphim Tsipelis)

Solution [by r9m]

Zero function

The following is an exercise proposed by one of our readers.

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $f\left( \frac{m}{2^n} \right)=0 \; \text{forall} \; m \in \mathbb{Z} \; \text{and} \; n \in \mathbb{N}$. Prove that $f=0$ forall $x \in \mathbb{R}$.

Solution

Integral with logarithm

Evaluate the integral:

$$\mathcal{J}=\int_0^1 \frac{1-x}{(x+1) \log x}\, {\rm d}x$$

Solution