What I see when I look at "Maths Item of the Month" is

Each of those, for where the denominators are a powers of the same number is a geometric sequence.

In particular, the sum when r= 1/2 is (1/4)(1/(1- 1/2))= 1/2. When r= 1/3, it is (1/9)(1/(1- 1/3))= 1/6. So this sum reduces to

That means, for example that , [tex]\frac{1}{3}= \frac{1}{2}- \frac{1}{3}[\math], etc.

In other words, this is a "telescoping" series. The second term in each is canceled by the first term in a second part of the sum.