Prove that the following function is continuous.

Question

I need to prove that the function: $$f(x)=\tan{x}+\frac{\tan^2 x}{2^2}+\frac{\tan^3 x}{3^2}+\cdots+\frac{\tan^n x}{n^2}+\cdots$$ is continuous for $x \in \left[-\dfrac{\pi}{4}, \dfrac{\pi}{4} \right]$.

I don't really know where to begin. I see, that for $-\dfrac{\pi}{4}$ and $ \dfrac{\pi}{4}$ it is a sum of a certain series, but don't know how to show the continuity.

Answer 1

The sequence of partial sums is a uniformly convergent (hint: $\|\tan\|_{\infty} =1$) sequence of continuous functions. So its limit, $f$, must be continuous.

Answer 2

Use what you know (that for the values you specified, it is a sum of a "certain series") together with the Weierstrass M-test.