Simplify using Geometric series

Question

Can this be simplified using Geometric series?

$$\Large\sum_{a=2}^\infty x^a\left[2(pq)^{\frac{a-2}2}+p^2+q^2\right]$$

thanks!

Answer 1

First split it into two summations:

$$\Large\sum_{a=2}^\infty 2\sqrt{pq}^{a-2}x^a+(p^2+q^2)\sum_{a=2}^\infty x^a\;.$$

The second summation is a simple geometric series, and the first can be rewritten as

$$\large\frac2{pq}\sum_{a=2}^\infty\left(\sqrt{pq}x\right)^a\;,$$

another geometric series.

Answer 2

Yes, you can solve $\sum_a x^a$ and $\sum_a (x\sqrt{pq})^a$ in terms of geometric series, which will give you your series. Note you have to check to make sure the series converges.