Mathematical Series: Formula & Concept

Instructor: Kimberlee Davison

Kim has a Ph.D. in Education and has taught math courses at four colleges, in addition to teaching math to K-12 students in a variety of settings.

In this lesson, you will learn the difference between a mathematical sequence and a mathematical series. You will also learn a little about summation notation.


A mathematical series is the sum of a list of numbers that are generating according to some pattern or rule. For example, '1+3+5+7+9' is a mathematical series - the sum of the first five odd numbers.

Sequence Versus Series

In mathematics, a set of numbers following some pattern is called a sequence if the numbers are simply listed with commas between them, such as the sequence of a perfect square below:

1, 4, 9, 16, 25,…

If you put a '+' sign between the numbers; however, then they become a series, as follows:


For example, suppose that your roommate baked an apple pie and left it out on the counter while she went off to work. Being very fair, you decide to only eat ½ the pie while she is gone; however, an hour later, you are still hungry, so you eat ½ of what is now left, or ¼ of the total pie.

Unfortunately for your roommate, her pie is delicious, so you find yourself every hour again eating half of what is remaining.

As a fraction of the total pie, your pie-eating during the eight hours she is at work looks like this:

½, ¼, 1/8, 1/16, 1/32, 1/64, 1/128, 1/256

Those last slivers of pie were pretty tiny, but you managed.

The list of fractions of pie is a sequence - it is simply a list with commas between each number.

If you want to know how much of the pie you ate altogether, then you create a series (a sum), that looks like this:

½ + ¼ + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/256

By the time your roommate returns, you have polished off 255/256 of the pie.

Another way you might write this is using summation notation, like this:

Sum of first 8 terms in a series

This notation says to take the number '1' and put it in for i in the expression to the right of the summation symbol (the big funny-looking symbol - the Greek letter 'sigma'). This gives you ½. Then you go to the next counting number, 2, and put it in place of the i. That gives you 1/(2^2), or ¼. You continue on until you have put in the counting number at the top of the summation symbol, 8. Then, add all those results together (sum them).

Finite and Infinite Sums

In the pie example, you only eat pie eight times, then you stop - your roommate comes home and claims those last few crumbs for herself. Since the series has an end, it is a finite series.

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