By: Tao Steven Zheng (鄭濤)

【Problem】

This problem is one of my favourite problems recorded in one of my high school mathematical notebooks!

Show that if , and are constants, then

【Solution】

Break the series into two parts

When , the first part of the sum is a convergent infinite geometric series. The formula for calculating this series is well known:

Now consider the second part of the sum

Factor our :

It is known that from the convergent infinite geometric series that

Notice that by differentiating the left-hand side of the above series, we get

Consequently,

and

Therefore,