Math Insight

Definition of the summation symbol

The symbol `\sum` indicates summation and is used as a shorthand notation for the sum of terms that follow a pattern.

For example, the sum of the first 4 squared integers, `1^2+2^2+3^2+4^2,` follows a simple pattern: each term is of the form `i^2,` and we add up values from `i=1` to `i=4.` We can write the sum compactly with summation notation as \[ \sum_{i=1}^4 i^2 = 1^2+2^2+3^2+4^2 = 20. \] Similarly, \[ \sum_{i=2}^6 3i = 3 \cdot 2 + 3 \cdot 3 + 3 \cdot 4 + 3 \cdot 5 + 3 \cdot 6 = 60. \] We don't have to use $i$ for the index, we could use another variable, like $j$: \begin{align*} \sum_{j=-2}^2 \frac{1}{j+3} &= \frac{1}{-2+3} + \frac{1}{-1+3} + \frac{1}{0+3} + \frac{1}{1+3} + \frac{1}{2+3}\\ &= 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} \end{align*}

In general, we define the sum as: \[ \sum_{i=a}^b f(i) = f(a) + f(a+1) + f(a+2) + \cdots + f(b-1) + f(b).\]