I'm working on a problem set for my partial differential equations class. We are studying convergence of Fourier Series. One of the questions is asking me to compute whether $$\sum_{n=0}^\infty (-1)^n x^{2n}$$ is pointwise convergent on the interval $-1<x<1$. The problem gives the hint that the partial sums can be explicitly computed. For some reason, my brain is emitting noxious levels of fart, and I can't find a formula for the partial sums of the series.
I have, however, managed to evaluate the series on the interval $-1<x<1$ without using partial sums by taking advantage of the fact that $$\sum_{n=0}^\infty (-1)^n x^{2n}=1-x^2\sum_{n=0}^\infty (-1)^n x^{2n}$$ and through rearranging, I found that $$\sum_{n=0}^\infty (-1)^n x^{2n}=\frac{1}{x^2+1}$$
If anyone would be willing to point me in the right direction to finding an explicit formula for the partial sums, that would be incredibly helpful. It has been a few years since I have taken Calculus II and I've clearly forgotten some things.