Summation

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:

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

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

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

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 $\lim _{n\to \infty }\sum _{k=1}^{n}k=+\infty$ or $\sum _{k=1}^{\infty }(-1)^{n}$), or convergent, meaning they equal a specific value (for instance,$\lim _{n\to \infty }\sum _{k=1}^{n}{\frac {1}{k^{2}}}={\frac {\pi ^{2}}{6}}$).

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

$\sum _{k=m}^{n}c\cdot f(k)=c\cdot \sum _{k=m}^{n}f(k)$

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:

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

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

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

If a sum is geometric, or in the form

$\sum _{k=0}^{n}ar^{k}={\frac {a(1-r^{n+1})}{1-r}}$

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:

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

See also

• Sequence
• Series
• Product operator