New posts in summation

Find the value $\binom {n}{0} + \binom{n}{4} + \binom{n}{8} + \cdots $, where $n$ is a positive integer.

summation power-series binomial-coefficients

Sum of reciprocals of binomial coefficients: $ \sum\limits_{k=0}^{n-1}\frac1{\binom{n}{k}(n-k)} $

combinatorics summation binomial-coefficients

Formalize a proof without words of the identity $(1 + 2 + \cdots + n)^2 = 1^3 + 2^3 + \cdots + n^3$

summation proof-writing

Prove $\sum_{i=0}^n (-1)^{n-i} \binom{n+1}{i} (i+1)^n = (n+2)^n$

calculus sequences-and-series combinatorics summation power-series

How find this $f(k)=\sum_{n=1}^{\infty}\frac{n^k}{2^n}$ is positive integers?

limits summation

Find the sum of $\binom{2016}{4} + \binom{2016}{8} +\binom{2016}{12} + \dots + \binom{2016}{2016}$

combinatorics summation binomial-coefficients

$\frac{1}{2^n}\binom{n}{n}+\frac{1}{2^{n+1}}\binom{n+1}{n}+...+\frac{1}{2^{2n}}\binom{2n}{n}=1$: short proof?

probability combinatorics summation combinations

The most peculiar totient sum: $\sum_{n=1}^{\infty} \frac{\phi(n)}{5^n +1}$

summation totient-function

Closed formula for the sums $\sum\limits_{1 \le i_1 < i_2 < \dots < i_k \le n} i_1 i_2 \cdots i_k $?

combinatorics summation generating-functions closed-form

Finite Series $\sum_{k=1}^{n-1}\frac1{1-\cos(\frac{2k\pi}{n})}$

real-analysis sequences-and-series trigonometry summation closed-form

Proving Nested Sum and Product Identities

sequences-and-series summation products

Evaluate $\int_{0}^{\pi} \frac{x\coth x-1}{x^2}dx$

calculus integration definite-integrals summation improper-integrals

Sum of series : $1+11+111+...$

algebra-precalculus summation arithmetic-progressions geometric-progressions

Evaluate $\frac{ 1 }{ 1010 \times 2016} + \frac{ 1 }{ 1012 \times 2014} + \frac{ 1 }{ 1014 \times 2012} + \cdots + \frac{ 1 }{ 2016 \times 1010} = ?$

sequences-and-series summation

Prove that $\sum\limits_{k=1}^n \frac{1}{k^2+3k+1}$ is bounded above by $\frac{13}{20}$

sequences-and-series algebra-precalculus inequality summation

For $n, m \ge 0, n+m\ge 2$, evaluate $W_{m, n} =\sum_{k=1}^{\infty} \frac{1}{(k+1)^mk^{n}} $.

summation recurrence-relations riemann-zeta

Proving that $\sum_{a=1}^{b} \frac{a \cdot a! \cdot \binom{b}{a}}{b^a} = b$

combinatorics summation induction binomial-coefficients

log of summation expression

summation logarithms exponential-sum

Find $\sum_{n=1}^\infty\frac{2^{f(n)}+2^{-f(n)}}{2^n}$, where $f(n)=\left[\sqrt n +\frac 12\right]$ denotes greatest integer function

sequences-and-series summation ceiling-and-floor-functions

Trigonometry limit's proof: $\lim_{x\to0}\frac{\sin(x)+\sin(2x)+\cdots+\sin(kx)}{x}=\frac{k(k+1)}{2}$

limits trigonometry summation limits-without-lhopital