summation formulas and properties

Given a sequence $a_{1}, a_{2}, ..., a_{n}$ of numbers, $n$ is non-negative integer

the finite sum $a_{1} + a_{2} + ... + a_{n}$ can be expressed as $\sum_{k = 1}^{n} a_{k}$

if n == 0, the value of the summation is defined to be 0.

the order in which the terms are added does not matter.

Given a finite sequences $a_{1}, a_{2}, ...$ of numbers,

we can write there finite-sum as $a_{1} + a_{2} + ...$ as $\sum_{k = 1}^{\infty} a_{k}$

diverges series, converges series

3 Linearity

real number c and any finite sequence $a_{1}, a_{2}, ..., a_{n}$ and $b_{1}, b_{2}, ..., b_{n}$

$\sum_{k = 1}^{n} (c a_{k} + b_{k}) = c \sum_{k = 1}^{n} a_{k} + \sum_{k = 1}^{n} b_{k}$

Arithmetic series

the summation

$\sum_{k = 1}^{n} k = 1 + 2 + ... + n$

is an arithmetic series and has the value

$\sum_{k = 1}^{n} k = \frac{n ( n + 1 )}{2}$

a general arithmetic series includes a additive constant a >= 0 and a constant b > 0 in each term, but has the same total asymptotically.

$\sum_{k = 1}^{n} (a + bk) = θ (n^{2})$

Squares $\sum_{k = 1}^{n} k^{2} = \frac{n ( n + 1 ) ( 2 n _{1} )}{6}$

Cubes

$\sum_{k = 1}^{n} k^{3} = \frac{n ^{2} ( n + 1 ) ^{2}}{4}$

real x != 1, the summation

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

the infinite decreasing geometric series occurs when the summation is infinite and |x| < 1

$\sum_{k = 1}^{\infty} x^{k} = \frac{1}{1 - x}$

if we assume that $0^{0} = 1$ , these formulas apply even when x = 0.