This post is inspired by a question in this year’s University of Melbourne Maths Olympics. We wish to find sums of the following form.
More generally, how do we find
The first sum is familiar enough to me – writing as the sum telescopes:
To do the next sum I thought one could use the partial fraction expansion
and continue as in the previous case. While this is doable, it is not easily generalisable. It is simpler to write
In the general case, we write
The reason for the title of the post is this sum can also be written as
Multiplying both sides by we also obtain the following interesting infinite sum involving reciprocals of binomial coefficients: