Series with Euler's totient function

Let $\varphi$ denote Euler's totient function. Evaluate the series:

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

Solution

A sinh integral

Let $s>1$. Evaluate the integral:

$$\mathcal{J}=\int_0^\infty \frac{x^{s-1}}{\sinh x} \, {\rm d}x$$

Solution

An Euler trigonometric sum

Prove that:

$$\sum_{n=1}^{\infty} \frac{\mathcal{H}_n}{n} \cos \frac{n \pi}{3}= -\frac{\pi^2}{36}$$

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

A digamma series

Let $\psi$ denote the digamma function and $\phi$ the golden ration. Prove that:

$$\sum_{n=1}^{\infty}\frac{\psi(n+\phi) - \psi \left ( n- \frac{1}{\phi} \right )}{n^2+n-1}= \frac{\pi^2}{2\sqrt{5}} + \frac{\pi^2}{\sqrt{5}} \tan^2 \left ( \frac{\pi \sqrt{5}}{2} \right )+ \frac{4\pi }{5}\tan \frac{\pi \sqrt{5}}{2}$$

(I. Mezo, China)

Solution (Ramya Dutta/ India)

The consequences of π

Prove that:

$$\sum_{j=2}^\infty \prod_{k=1}^j \frac{2 k}{j+k-1} = \pi$$

Solution

Sequence of zero limit

Let $\{a_n\}_{n \in \mathbb{N}}$ be a sequence of real number such that the series $\sum \limits_{n=1}^{\infty} \frac{a_n}{n}$ converges. Prove that:

$$\lim_{n \rightarrow +\infty} \frac{a_1+a_2+\cdots +a_n}{n}=0$$

Solution

Nested sequence and series

Let $a_n =\underbrace{\sin \left ( \sin \cdots \left ( \sin x \right )\cdots \right )}_{n \;\; {\rm times}}$. Examine if the series $\sum \limits_{n=1}^{\infty} a_n$ converges.

Solution

Sequence of series

Let $f_n: \mathbb{R} \rightarrow \mathbb{R}$ be defined as:

$$f_n(x)=\frac{\cos nx}{n^2}, \; \; n \in \mathbb{N}$$

Prove that the series $\displaystyle \sum_{n=1}^{\infty} f_n$ converges uniformly to a function $f:\mathbb{R} \rightarrow \mathbb{R}$ and that:

$$\int_0^{\pi/2} f(t)\, {\rm d}t = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^3}$$

Solution

Ramanujan's Result

Prove that:

$$\sum_{k=1}^{\infty}\frac{1}{k^3 (k+1)^3}= 10 - \pi^2$$

Solution

Series and inequality

Prove that:

$$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n} (n+1)}<2$$

IMC 2015 / Round 2 Problem 1

Solution

Evaluating series using digamma

Evaluate the series:

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

Solution

Series involving arctan

Evaluate the series:

$$\sum_{n=1}^{\infty} \arctan \frac{1}{n^{2}} $$

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

Limit of a sequence

Let $(a_n)_{n\in \mathbb{N}}$ be a sequence of real numbers such that the series $\displaystyle \sum_{n=1}^{\infty} \frac{a_n}{n^s} ,\; s>0$ converges. Prove that:

$$\lim_{n \rightarrow +\infty} \frac{a_1+a_2+\cdots+a_n}{n^s}=0$$

Solution:

Series converges

Let $\mathbb{P}$ denote the set of prime numbers. Then prove that the series $\displaystyle \sum_{\mathbb{P}}\frac{1}{p \ln p} $ converges, where $ p \in \mathbb{P}$.

Solution:

Evaluating series $\sum_{n=0}^{\infty}\frac{n}{2^{n+1}\left ( n+1 \right )^2}$.

Evaluate the series:

$$\sum_{n=0}^{\infty}\frac{n}{2^{n+1}\left ( n+1 \right )^2}$$

Solution:

Series

Evaluate the series:
$$\mathcal{S}=\sum_{n=1}^{\infty}\frac{2n+1}{2^n}$$

Solution:

Series (alternating series)

Prove the series:

$$\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^3}=\frac{\pi^3}{32}$$

Solution:

$\sum_{n=0}^{\infty}\left ( \frac{2}{27} \right )^n \binom{3n}{n}$

Evaluate the series:
$$\sum_{n=0}^{\infty}\left ( \frac{2}{27} \right )^n \binom{3n}{n}$$

Answer: $\displaystyle \frac{\sqrt{3}+1}{2}$.

Proof: