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Number Theory

Summation Notation Basics

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Summation of sequence from X=1 to N of $X^2$

$\frac{N(N+1)(2N+1)}{6}$

Summation of sequence from X=1 to N of $3X^2$

$\frac{N(N+1)(2N+1)}{2}$

Summation of sequence from X=1 to N of $4X^3$

$2N^2(N+1)^2$

Summation of sequence from X=1 to N of $X(X-1)$

$\frac{N(N-1)(N+1)}{3}$

Summation of sequence from X=1 to N of $X(X+1)$

$\frac{N(N+1)(N+2)}{3}$

Summation of sequence from X=1 to N of 1

$\frac{N(N+1)}{2}$

Summation of sequence from X=1 to N of X

$\frac{N(N+1)}{2}$

Summation of sequence from X=1 to N of $\log(X)$

No simple closed-form formula

Summation of sequence from X=1 to N of $X^3$

$\left( \frac{N(N+1)}{2} \right)^2$

Summation of sequence from X=1 to N of $\sin(X)$

No simple closed-form formula

Summation of sequence from X=1 to N of $(-1)^{X+1}$

$\frac{1 + (-1)^N}{2}$

Summation of sequence from X=1 to N of $\cos(X)$

No simple closed-form formula

Summation of sequence from X=1 to N of $2^{-X}$

$1 - 2^{-N}$

Summation of sequence from X=1 to N of $5$

$5N$

Summation of sequence from X=1 to N of $2X+1$

$N^2+N$

Summation of sequence from X=1 to N of $\sqrt{X}$

No simple closed-form formula

Summation of sequence from X=1 to N of $2^X$

$2^{N+1} - 2$

Summation of sequence from X=1 to N of $2X$

$N(N+1)$

Summation of sequence from X=1 to N of $\frac{1}{X}$ (Harmonic Series)

No simple closed-form formula

Summation of sequence from X=1 to N of $X!$ (Factorial Sum)

No simple closed-form formula

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