Geometric series in closed form.

Question

$$f=\sum_{k=3}^{\infty}(-1)^kx^{(2k-2)}$$

i would like to write this as the geometric power series!

Is there a ritual you have to do to solve this? thanks in advance.

You may use the fact that $(-1)^k x^{2k-2} = -\left(-x^2\right)^{k-1}$. — Joel Cohen, Jun 22 '15 at 13:02
See http://math.stackexchange.com/questions/1333715/solving-sum-of-1n-1-2n/1333719#1333719 — lab bhattacharjee, Jun 22 '15 at 13:11

score 5 · Accepted Answer · answered Jun 22 '15 at 13:02

5

$$\sum_{k\geq3}\left(-1\right)^{k}x^{2k-2}=\frac{1}{x^{2}}\sum_{k\geq3}\left(-x^{2}\right)^{k}=-x^{4}\sum_{k\geq0}\left(-x^{2}\right)^{k}. $$

answered Jun 22 '15 at 13:02

Marco Cantarini

1 Answers1