Is this a power series? Is the following a power series?
$$\sum_{n=0}^\infty a_k \left( \frac{2x}{1+x^2} \right)^k \ , x \in (-1,1)$$
where $a_k$ is a bounded sequence. 
I was asked to show that this power series converges, and that the sum function was continuous. 
Two things confused me: Why are they calling this weird-looking thing a "power series"? And why are they asking for "convergence"? Isn't it customary to ask for either pointwise or uniform convergence, especially with the question of continuous sum function in mind?

It's a power series in $u = \frac{2x}{1+x^2}$.
As such, if $|u| < 1$ for $x \in (-1, 1)$, then as long as $a_k$ is bounded, the series is convergent.
Fortunately, we can indeed show that $|u| < 1$.  When $0 \leq x < 1$,
$$
0 < (x-1)^2 \leq 1
$$
$$
0 < x^2-2x+1 \leq 1
$$
$$
-1-x^2 < -2x < -x^2
$$
$$
1+x^2 > 2x > x^2 \geq 0
$$
$$
0 \leq \frac{2x}{1+x^2} < 1
$$
Complete similar reasoning for $-1 < x \leq 0$, and you are done.
