According to a proof in a textbook, the order of summation of the following series

\displaystyle \sum^{\infty}_{n=1} \sum^{\infty}_{k=1} (-1)^{k+1}\frac{x^{2k}}{kn^{2k}}

can be interchanged because they are absolutely convergent. Why is this the case? And how do I go about proving that the series is absolutely convergent?

Thanks in advance,

HTale