Summation notation

Summation notation (or sigma notation) allows us to write a long sum in a single expression.

This is the sigma symbol: $\displaystyle\sum$sum. It tells us that we are summing something.

Let's start with a basic example:

This is a summation of the expression $2n-1$2, n, minus, 1 for integer values of $n$n from $1$1 to $3$3:

Notice how we substituted $\goldD{n=1}$start color #e07d10, n, equals, 1, end color #e07d10, $\goldD{n=2}$start color #e07d10, n, equals, 2, end color #e07d10, and $\goldD{n=3}$start color #e07d10, n, equals, 3, end color #e07d10 into $2\goldD n-1$2, start color #e07d10, n, end color #e07d10, minus, 1 and summed the resulting terms.

$n$n is our summation index. When we evaluate a summation expression, we keep substituting different values for our index.

We can start and end the summation at any value of $n$n. For example, this sum takes integer values of $n$n from $4$4 to $6$6:

We can use any letter we want for our index. For example, this expression has $i$i for its index:

Some summation expressions have variables other than the index. Consider this sum:

$\displaystyle\sum_{n=1}^4 \dfrac{k}{n+1}$sum, start subscript, n, equals, 1, end subscript, start superscript, 4, end superscript, start fraction, k, divided by, n, plus, 1, end fraction.

Notice that our index is $n$n, not $k$k. This means we substitute the values into $n$n, and $k$k remains unknown:

Key takeaway: Before evaluating a sum in summation notation, always make sure you identified the index, and that you are only substituting into that index. Other unknowns should remain as they are.