Sum of Infinite Geometric Sequence

Hey math friends and happy Wednesday! Today we are going to take a look at how to find the sum of infinite geometric sequence (aka series) in summation notation. This may sound complicated, but lucky for us there is an already existing formula that is ready and easy for us to use! So, let’s get to it! Also, don’t forget to check out the video and practice problems below for even more. Happy calculating! ?

Looking to review different types of sequences? Check out this post here!

What does it mean to find the “Sum of Infinite Geometric Sequence” (Series)?

We already know what a geometric sequence is: a sequence of numbers that forms a pattern when the same number is multiplied or divided to each term.

Example:

But when what happens if we wanted to sum all the terms of our geometric sequence together?

Example:

We can also write our infinite geometric series using Summation Notation:

How would we calculate that?  That’s where our Infinite Geometric Series formula will come in handy! But before we find a solution, lets take a closer look at what geometric series qualify for this formula.

Divergence Vs. Convergence:

There are two types of Infinite Geometric Series:

Type 1: Infinite Geometric Series that diverges to infinity

Type 2: Infinite Geometric Series that converges to a numeric value (-1 < r < 1)

Check out the differences in the example below:

Infinite Geometric Series Formula:

Note that the below infinite geometric series formula can only be used if the common ratio, r, is less than 1 and greater than -1. If the common ratio, r, is not between -1 and 1, then the sum of the geometric sequence diverges to infinity (and the formula cannot be used).

Now that we have a formula to work with and know when to use it (when -1 < r < 1), let’s take another look at our question and apply our infinite geometric series formula to find a solution:

Step 1: First, let’s identify the common ratio to make sure that its between -1 and 1.

Step 2: Now that we know we can use our formula, let’s write out each part and identify what numbers we are going to plug in.

Step 3: Now let’s fill in our formula and solve with the given values.

Practice Questions:

State if each series converges or diverges, then if applicable find the solution.

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! ?

*Also, if you want to check out Finite Geometric Series click this link here!

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