Another Fibonacci series

Let $F_n$ denote the $n$ -th Fibonacci number. Evaluate the sum:

$$\mathcal{S} = \sum_{n=0}^\infty \sum_{k=0}^n \frac{F_{2k}F_{n-k}}{10^n}$$

Solution

A series involving Fibanacci

Let $F_n$ denote the $n$-th Fibonacci number with initial values $F_1=F_2=1$. Prove that:

$$\sum_{n=0}^{\infty} \arctan \frac{1}{F_{2n+1}} = \frac{\pi}{2}$$

Solution

An equality of two finite sums

Let $n \in \mathbb{N}$. Prove that:

$$\sum_{k=1}^{n}\frac{1}{4k^{2}-2k}=\sum_{k=n+1}^{2n}\frac{1}{k}$$

Solution

A Fibonacci sum

Let $F_n$ denote the $n$-th Fibonacci sum. Evaluate (in a closed form) the sum:

$$\sum_{n=0}^{N} \frac{1}{F_{2^n}}$$

Solution

Finite sum

Prove that:

$$\sum_{k=0}^{n} \frac{(-1)^k}{k+1}\binom{n}{k}=\frac{1}{n+1}$$

Solution

Trigonometric treat

Evaluate the sum:

$$\frac{1}{\cos 0^\circ \cos 1^\circ}+\frac{1}{\cos 1^\circ \cos 2^\circ}+ \frac{1}{\cos 2^\circ \cos 3^\circ}+\cdots+ \frac{1}{\cos 88^\circ \cos 89^\circ}$$

Solution

Sum with complex numbers

Let $z \in \mathbb{C}$ such that $z^{2017}=1$ and $z\neq 1$. Evaluate the sum $\displaystyle \sum_{n=1}^{2017} \frac{1}{1+z^n}$.

Solution

Sum of cosines

Show that:

$$\cos \frac{\pi}{5} + \cos \frac{3\pi}{5}= \frac{1}{2}$$

Solution

Finite sum

Evaluate the sum:

$$S= \frac{1}{1+2}+ \frac{1}{1+2+3}+\cdots + \frac{1}{1+2+3+\cdots +2015}$$

Solution

Alternate binomial sum

Prove that:

$$\sum_{k=0}^{2n}(-1)^k \binom{2n}{k}^2 = (-1)^n \binom{2n}{n}$$

Solution

Alternating Euler Sum

Evaluate the sum:

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

Solution: