what is the sum of $\sum_{k=1}^T x^{2k}$?

Question

what is the sum for the following series $$ x^{2\cdot 1} + x^{2\cdot 2} + x^{2\cdot 3} \ldots + x^{2\cdot T} $$

lower and upper bounds are also welcome.

Answer 1

This is a geometric series. You have $$ \sum_{k=1}^T x^{2k}=\sum_{k=1}^T (x^2)^k=\frac{x^2-(x^2)^{T+1}}{1-x^2}=\frac{x^2-x^{2T+2}}{1-x^2}. $$