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{{ printedBook.courseTrack.name }} {{ printedBook.name }} The **summation notation**, also known as *sigma notation*, allows long sums to be written in a compact way. It is denoted using the Greek letter sigma $(Σ),$ along with several other pieces of information about the summation.

The variable $n$ is called the *summation index* and only takes integer values. To write this sum explicitly, the variable $n$ appearing after sigma must be substituted with the integers from the initial value through the last value. $n=1∑4 2nn=1∑4 2nn=1∑4 2n =n=12(1 )+n=22(2 )+n=32(3 )+n=42(4 )=2+4+6+8=20 $
Notice some aspects of summation notation:

- The summation does
**not**depend on the summation index used. $n=1∑4 2n=k=1∑4 2k=j=1∑4 2j=20 $ - Sometimes a summation may involve other variables. These should not be confused with the summation index. $n=1∑3 k2n $ Here, the summation index is $n.$ Therefore, the indicated values should only be substituted into $n$ and not into $k.$ $n=1∑3 k2n n=1∑3 k2n n=1∑3 k2n =n=1k2(1) +n=2k2(2) +n=3k2(3) =k2 +k4 +k6 =k20 $
- The initial value can be any integer less than or equal to the number above the summation symbol. The number above $Σ$ only indicates the final value needed to be substituted for $n,$
**not**the number of terms. $n=8∑10 3n =n=83(8) +n=93(9) +n=103(10) $

The summation notation is not only useful for working with sums involving a large number of terms, but it can also be used to represent an *infinite sum*.
$n=1∑∞ 2_{n}1 =2_{1}1 +2_{2}1 +2_{3}1 +… $