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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.

See also

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