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3.7: Series

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    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;
    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.

    This page titled 3.7: Series is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Allen B. Downey (Green Tea Press) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.