Infinite Series: Applications, Formula & Examples

Instructor: David Karsner

David holds a Master of Arts in Education

An infinite series, represented by the capital letter sigma, is the operation of adding an infinite number of terms together. This summation will either converge to a limit or diverge to infinity.

Karl is training to run a marathon race; however, he has an unusual training plan. On the first day of training, he runs a mile. The next day, he runs 1/2 of a mile, and the day after that he runs 1/4 mile more. Each subsequent day he runs half of what he ran the day before. If he keeps this training up forever, how many miles will he have run? Although this training is an unusual way to train for a marathon, it does illustrate a situation that would require the use of an infinite series. This lesson will illustrate the use of infinite series and give examples of common series as well as their applications.

Summation Notation
Sigma Notation

To find the distance Karl ran, we would add 1 mile + 1/2 mile + 1/4 mile + 1/8 mile + 1/16 mile . . . . Notice how each term can be written as 1/(a power of two) or 1/(2^n). If n goes from 0 to infinity, then Karl's training can be represented with the Summation Notation image. After two days, Karl has run 1 1/2 miles. He has 1/2 mile left to run before he reaches two miles. On day three, he runs 1/4 mile for a total of 1 3/4 miles, leaving 1/4 mile to run before reaching two miles. This pattern will continue so that he will run half each day of what is left before hitting two miles. He will never get completely to two miles because he is running half of what's left. So 2 miles is the limit of how far he will run.


An infinite series is the addition of an infinite number of terms. A lot of vocabulary is associated with infinite series and several of these have been listed below.

Summation Notation is represented by the capital Greek letter sigma. It denotes that all the items in the list will be added together.

Addends: the items in the sequence that are going to be added (in the Summation Notation image, the addends are 1/2^n).

Index: the smaller letter and number found below the sigma. It is very often represented with an n and tells which addends will be used. When adding the items in the series, you begin with the number below the sigma and continue adding till you reached the number above the sigma. With an infinite series, the number above the sigma will be the infinity symbol, meaning you keep on adding.

Sequence: a set of terms or numbers that have been placed in order.

Series : a sequence that has been added together. This sequence can be a limited number of terms, called a finite series, or have terms that continue to infinity, called an infinite series.

Partial Sum: when a finite (fixed) number of terms in an infinite series have been added together. In an infinite series, the partial sum will be approaching the limit of the series.

Finite Series : when a fixed (finite) number of terms are added together.

With an infinite series, the primary concern is that of convergence. Does the summation of infinite terms approach a set value? This set value would be its limit. If an infinite series has a limit, then it is a convergent series. If it does not, it is a divergent series. A series will be convergent if the addends when n is very large are equivalent to zero. An infinite series is also convergent to a limit L if the summation of the partial sum of that same series is equal to the same limit L. We can test for convergence in many ways: n-th term test, comparison test, ratio test, and Cauchy condensation test are a few of those.


We use many different kinds of infinite series. The 'Types of Infinite Series' image gives several of the more commonly used infinite series.

Common Types of Infinite Series

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