Power series with $x^{4n}$

Question

I'm new to this Forum. I do not find an approach to solve the following problem (from the book "Herbert Wallner, Aufgabensammlung Mathematik Band 1", so this is not a homework question):

For which $x\in\mathbb{R}$ does the series $$\sum_{n=1}^{\infty}\frac{x^{4n}}{(2+3x^{4})^{n+1}}$$ converge.

According to the book the solution is: For $x\in\Bbb{R}$.

I think at first I have to write the series in the form $\sum_{n=n_0}^{\infty}a_n(x-x_0)^n$, so that I can apply the other methods for power series examination (radius of convergence etc), but I do not know how.

Answer 1

Hint

Note that

$$0\le \frac{x^{4n}}{(2+3x^{4})^{n+1}}=\frac{1}{2+3x^4}\left(\frac{x^4}{2+3x^4} \right)^n< \frac{1}{2+3x^4}\frac{1}{3^n}\le \frac{1}{2\cdot3^n}$$

$\forall x\in \mathbb{R},\forall n\in \mathbb{N}.$