How to Sum an Infinite Series

Comparing accurate theory and computational estimates

I recently took on three interesting problems of increasing difficulty related to calculating the sums of infinite series. After I had found solutions to each part, I then went back and discovered a different approach to solving them which delivered the same result. The process of finding these alternative methods led to some realizations about the relationship between infinite sums and well-known algebraic expansions of functions. Finally, I tested each solution by designing an algorithmic approximation in some programming languages, which verified that my solutions were accurate. Here are the three series:

This was so much fun if you are mathematically minded, so I wanted to share the problems and my solutions with you. Let’s start with the easier of the three.

Series (i) Method 1

Series (i) is the following:

The first method I used to solve this was to notice that this series is actually a sum of an infinite number of infinite geometric series. The first part of the sum is the standard geometric series: