I have numerically verified that the following sum comes to , the same thing the back of the book says. So what'd I do wrong?

Find the sum of the convergent series

.

I verified that the above sum can be represented as

,

which can be rewritten as

.

I believe I can apply the geometric series to the summation, giving me

.

Ahhh... but now I see that the geometric series equation starts the summation at (do you find that typing up your answers sometimes helps you realize what's wrong? Why is that?).

So, I need to subtract from my answer, which (unsurprisingly) gives .