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 numbers from 1 to 5 can be written as:

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

Any operation can be performed on i. For instance,

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

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

Properties

A sum of any series in which i 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,

$\sum _{i\mathop {=} m}^{n}Ci=C\sum _{i\mathop {=} m}^{n}i$

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.

$\sum _{i\mathop {=} m}^{n}(i+i)=\sum _{i\mathop {=} m}^{n}i+\sum _{i\mathop {=} m}^{n}i$

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

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

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

$\sum _{i\mathop {=} 1}^{n}(i^{3})={\frac {n^{2}(n+1)^{2}}{4}}$

If a sum is geometric, or in the form

$\sum _{i\mathop {=} 0}^{n}(ar^{i})$,

the sum is equal to

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

If the sum of a geometric series is infinite and convergent ($-1<r<1$), 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.