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} $
