3.7: Series

In mathematics, a series is the sum of the elements of a sequence. It’s a terrible name, because in common English, “sequence” and “series” mean pretty much the same thing, but in math, a sequence is a set of numbers, and a series is an expression (a sum) that has a single value. In math notation, a series is often written using the summation symbol $\sum$.

For example, the sum of the first 10 elements of $A$ is $$\sum_{i=1}^{10} A_i\notag$$

A for loop is a natural way to compute the value of this series:

Listing 3.1: A program that calculates a simple series

A1 = 1;
total = 0;
for i=1:10
    a = A1 * (1/2)^(i-1);
    total = total + a;
end
ans = total

Let’s walk through what’s happening here. A1 is the first element of the sequence, so we assign it to be 1; we also create total, which will store the cumulative sum. Each time through the loop, we set a to the $i$th element and add a to the total. At the end, outside the loop, we store total as ans

The way we’re using total is called an accumulator, that is, a variable that accumulates the result a little bit at a time.

Exercise $3.4$

This example computes the terms of the series directly. As an exercise, write a script named series.m that computes the same sum by computing the elements recurrently. You will have to be careful about where you start and stop the loop.