Summation symbol

Elementary combinatorics: Summation and product symbol

Summation symbol

A small number of mathematical terms can be added easily with the sum symbol $+$ . But in many problems there are summations in which many terms have to be added together. For example, the sum of the natural numbers 1 up to and including 64, or the sum of the first 8 squares. You can of course write this suggestively as $1+2+3+\ldots+63+64=2080$ and $1+4+16+\ldots+47+64=204$ But we have included many terms in the expression here for a reason. With $1+2+\ldots+64$ it is already unclear whether you mean the sum of the numbers 1 up to and including 64, or actually the sum $1+2+4+8+16+32+64=126$ is meant, where each term is twice the previous term. Of course, the notation $1+4+\ldots 64$ does not have to indicate the sum of the first 8 squares, but it could also stand for $1+4+16+64=85$ where each term is four times the previous term. If you find these examples a bit contrived, then what do you think $3+5+7+\ldots 17$ means? The sum of odd numbers $3+5+7+9+11+13+15+17$ or the sum of prime numbers $3+5+7+11+13+17\text?$

In mathematics you need a more precise sum notation than `dotting' and that is sigma notation. The symbol $\sum$ , which is pronounced ``sigma'', derived from the Greek capital letter $\Sigma$ , is used to indicate addition of terms; it is summation symbol. A so-called summation index runs over the various terms: the sum of the natural numbers 1 to 64 then becomes $\displaystyle\sum_{i=1}^{64} i$ . We read this as ``the sum of $i$ where $i$ runs from $1$ through $64$ '' or as ``the sum of $i$ for $i=1$ through $i=64$ ''. The pattern is now clearly written down and $\displaystyle\sum_{i=1}^{66} i=2080$ . The sum of the first 8 squares is denoted $\displaystyle\sum_{i=1}^{8} i^2$ in sigma notation and is equal to $204$ because ``the sum of $i^2$ for $i=1$ through $i=8$ '' does indeed equal this number.

The sum $2+3+4+5+6+7+8+9+10$ can be shortened as $\sum_{i=2}^{10} i,$ but also as $\sum_{j=1}^{9} (j+1)$ or as $\sum_{j=4}^{12} (j-2)\text.$ The sigma notation for a certain sum is therefore not unique.

If we have 10 numbers that we denote with $a_0,a_1,a_2,\ldots, a_9$ , then we write the sum of these numbers as $\sum_{i=0}^{9} a_i=a_0+a_1+a_2+\cdots + a_8+a_9$ We read this as ``the sum of the terms $a_i$ where (the summation index) $i$ goes from $0$ up to and including $9$ ''.

For each natural number $n$ and $a_0, a_1,\ldots,, a_n$ mathematical objects for which addition is defined (e.g., real numbers) we define: $\sum_{i=0}^n a_i=a_0+a_1+a_2+\ldots a_n$ More generally for natural numbers $m\le n$ : $\sum_{i=m}^n a_i=a_m+a_{m+1}+a_{m+2}+\ldots a_n$ We read this as ``the sum of the terms $a_i$ where (the summation index) $i$ goes from $m$ up to and including $n$ ''. Here $a_i$ is the general term, $i$ is the summation index, $m$ is the lower bound, and $n$ is the upper bound of the summation.

Alternative notation Sometimes you come across in textbooks the expression $\displaystyle\sum_{i=m}^n a_i$ instead of $\sum\!\!\!\!\!\!\!\!\!\phantom{\sum}_{i=m}^n a_i$ .

letter choice Which summation index you use is your own choice: $\displaystyle\sum_{i=m}^n a_i=\sum_{k=m}^n a_k$ .