A calculus of the absurd

Chapter 3 Sequences and series

3.1 Sigma notation

One way of writing long sums (commonly used at GCSE), is to use an elipsis (the $...$ symbol). For example, if we were to write the sum of all the positive integers from $1$ to $21$ we could write it out in full as

\begin{align} & 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 \\ \notag & + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 \end{align}

The other way we could write it is as

\begin{equation} 1 + 2 + 3 + ... + 21 \end{equation}

Here we’ve used $...$ to stand for all the terms between 2 and 21. This notation works, but there’s another way we could write this sum; using sigma notation! In this case, we could write this as

\begin{equation} \sum _{i=1}^{21} i \end{equation}

Sigma notation is not as bad as it looks! All this means (when read aloud) is “the sum of all the values of $i$ where $i$ starts at $1$ and ends at $21$ (inclusive¹¹ i.e. we include $1$ and $21$)”.

In general there are three main parts to sigma notation - the place where we start counting from, the place where we finish counting and the "variable of indexation". In a more general case, we would have something of the form

\begin{equation} \sum _{i=0}^{n} \left [ \text {some expression depending on $i$} \right ] \end{equation}

This would mean that we start at $i=0$ and find the value of whatever the expression depending on $i$ is. Say, for the sake of example, it happened to be²² This is just an example - the expression could be anything! $3i^2$. In this case, we would have $3 \cdot 0 ^2 = 0$. We would then add this to the value of the expression at $1$ ($3 \cdot 1^2$), at $2$ ($3 \cdot 2^2$), at $3$ ($3 \cdot 3^2$), and so on (all the way to $n$ - $3 \cdot n^2$).