Series convergence for every real number

Question

We have the sequence $$x_n=\sum_{i=1}^n \frac{(2\alpha)^i}{4^i+(\alpha^2)^i}$$ and we have to prove that it is convergent $\forall \alpha \in \mathbb{R}$. If we use the Absolute Convergence Theorem, we only have to prove it for $\alpha \gt 0$. The sequence is clearly monotonous increasing, so how can I find an upper bound?

Answer 1

HINT:

$$\left|\frac{(2\alpha)^i}{4^i+\alpha^{2i}}\right|\le \left|\frac{(2\alpha)^i}{4^i}\right|$$

and

$$\left|\frac{(2\alpha)^i}{4^i+\alpha^{2i}}\right|\le \left|\frac{(2\alpha)^i}{\alpha^{2i}}\right|$$

But what happens when $|\alpha|=2$?