Summation

Symbol of sum

Notation of sum

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 integers 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 and is denoted

$\lim_{n\to\infty}\sum_{k=1}^n \equiv \sum_{k=1}^{\infty}$

Infinite sums can be divergent, meaning they do not converge (such as $\lim_{n\to\infty}\sum_{k=1}^nk=+\infty$ or $\sum_{k=1}^{\infty} (-1)^n$), 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 convergent sum of any series in which $f(k)$ is multiplied by a constant is the same as the entire sum multiplied or divided by said constant. If $c$ is a constant then

$\sum_{k=m}^nc\cdot f(k) =c\cdot\sum_{k=m}^n f(k)$

Two convergent series added together with the same index are equal to the series sum of the arguments:

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

Some example sums with closed forms are shown below:

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

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

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

If a sum is geometric, or in the form

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

If the sum of a geometric series is infinite and convergent $(|r|<1)$, the formula simplifies to:

$\sum_{k=0}^{\infty}ar^k=\frac{a}{1-r}$

If $|a|>1$, then:

$\sum_{k=1}^{\infty}{1\over a^k}={1\over a-1}$

Definitions and Examples

For any formula that diverges, that sum is going to be infinity. For example:

$\sum _{k=1}^{\infty }n=\infty$

For any fraction formula that goes to 0, then its sum is 1. For example:

$\sum _{k=1}^{\infty }{\frac {1}{2^{n}}}=1$

See also

• Sequence
• Series
• Product operator