## FANDOM

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Summation is the operation of adding a sequence of numbers to get a sum or total. It is usually denoted with the letter sigma ($\sum$). A sum of all the numbers from 1 to 5 can be written as:

$\sum_{k=1}^5k=1+2+3+4+5=15$

Any operation can be performed on $k$ . For instance,

$\sum_{k=1}^5k^3=1^3+2^3+3^3+4^3+5^3=1+8+27+64+125=225$

A partial sum, where the sum is only of part of a series, is also called a finite sum. If a sum is between a number and infinity, it is called a series. Infinite sums can be divergent, meaning they approach infinity (such as $\lim_{n\to\infty}\sum_{k=1}^nk=\infty$), or convergent, meaning they equal a specific value (for instance, $\lim_{n\to\infty}\sum_{k=1}^n\frac1{k^2}=\frac{\pi^2}{6}$).

## Properties

A sum of any series in which $k$ is multiplied or divided by a constant is the same as the entire sum multiplied or divided by said constant. If $c$ is a constant,

$\sum_{k=m}^nc\cdot k=c\cdot\sum_{k=m}^nk$

If the sequence is of two numbers added to each other, the answer will be the same as the sums of both terms added together.

$\sum_{k=m}^n(k+k)=\sum_{k=m}^nk+\sum_{k=m}^nk$

Sums where $k$ is raised to a power can be found with the following formulas:

$\sum_{k=1}^nk=\frac{n(n+1)}{2}$
$\sum_{k=1}^nk^2=\frac{n(n+1)(2n+1)}{6}$
$\sum_{k=1}^nk^3=\left(\frac{n(n+1)}{2}\right)^2=\left(\sum_{k=1}^nk\right)^2$

If a sum is geometric, or in the form

$\sum_{k=0}^nar^k$

the sum is equal to

$\frac{a(1-r^{n+1})}{1-r}$

If the sum of a geometric series is infinite and convergent ($|r|<1$), the formula still applies, but since

$\lim_{n\to\infty}\frac{a(1-r^{n+1})}{1-r}=\frac{a}{1-r}$

we can use the latter formula.