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]

Integral with dilogarithm

Let ${\rm Li}_2$ denote the dilogarithm function. Evaluate the integral

$$\int_0^\infty \frac{{\rm Li}_2(-x)}{1+x^2}\, {\rm d}x$$

Solution

A zeta series

Evaluate the series:

$$\sum_{n=1}^{\infty} \frac{\zeta(2n)-\zeta(3n)}{n}$$

Solution

A weird infinite product

Evaluate the following product

$$P =\left({\frac{2}{1}}\right)^{1/8}\cdot\left({\frac{{2\cdot 2}}{{1\cdot 3}}}\right)^{3/16}\cdot\left({\frac{{2\cdot 2\cdot 2\cdot 4}}{{1\cdot 3\cdot 3\cdot 3}}}\right)^{6/32}\cdot\left({\frac{{2\cdot 2\cdot 2\cdot 2\cdot 4\cdot 4\cdot 4\cdot 4}}{{1\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 5}}}\right)^{10/64}\cdots$$

Solution

Alternating binomial series

Evaluate the series:

$$\mathcal{S}=\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{4^n n^2} \binom{2n}{n}$$

Solution

Gautschi's Inequality for Gamma function

Prove that:

$$x^{1-s} < \frac{\Gamma(x+1)}{\Gamma(x+s)} < (x+1)^{1-s},\qquad x > 0,\; 0 < s < 1$$

which is better known as Gautschi's Inequality , due to Walter Gautschi.

Solution

Integral with digamma

Let $\psi(x)$ denote the digamma function. Evaluate the integral:

$$\int_0^1 \psi(x) \sin 2n \pi x \, {\rm d}x \quad , \qquad n \in \mathbb{N} $$

Solution

An integral with arctan

Evaluate the following integral:
$$\int_0^{2-\sqrt{3}}\frac{\arctan x}{x}\, {\rm d}x$$

The result is due to Ramanujan.

Solution

Evaluating series using digamma

Evaluate the series:

$$\sum _{n=0}^{\infty }\frac{1}{(4n+1)(4n+3)}$$

Solution

There does not exist function

Prove that there does not exist an elemenary function $f$ such that $f({\rm glog}x)$ is an antiderivative of ${\rm glog}x$.

(${\rm glog}x$ denotes the inverse function of $e^x/x$ and is called generalized logarithm.)

Solution

Series of Bessel function

Evaluate the series:

$$\sum_{n=1}^{\infty} \frac{J_0(2n)}{n^2}$$

where $J_0$ is the Bessel function of the first kind.

Solution

Transcedental number

Examine whether or not the number $\displaystyle \sum_{n=1}^{\infty} \frac{1}{2^{n^2}}$ is transcedental.

Solution

On polygamma reflection formula

In this thread we are proving the polygamma reflection formula stating that:

$$\psi^{(n)}(1-z)+(-1)^{n+1}\psi^{(n)}(z)=(-1)^n  \pi \frac{\mathrm{d}^n }{\mathrm{d} x^n}\cot \pi z$$