Irrational Exuberance

Submitted by Pingo from Lisboa, 3/15/2000. Original answer and this article by Allen Stenger.

Suppose n is a natural number and r a rational number, 0 < r < 1. Consider the series



Is it irrational?

Hint 1

In general it is almost impossible to tell from an infinite series whether the sum is rational or irrational. Instead, try to find a closed-form expression for this series.

Hint 2

Part of the general term can be factored out to get the binomial coefficient

Need another hint? Click here.

Click here for the complete solution.