# 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

$$1 + 2 + 3 + ... + 21$$

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

$$\sum _{i=1}^{21} i$$

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

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

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 $$n$$ - $$3 \cdot n^2$$).