Let $(a_n)_{n\in \mathbb{N}}$ be a bounded sequence. Prove that the series $\displaystyle \sum_{n=1}^{\infty} \frac{a_n}{n^{2x}}$ converges absolutely and uniformly on $(1, +\infty)$ to a differentiable function $f$.
The result that the series converges absolutely and uniformly comes from the M- test so there exists a function $f$ that the series converges uniformly. How that to prove that the function is differentiable? This where I am stuck.
Edit: I thought the last part of the question was something I was overseeing. However, I cannot proceed .. I have a feeling that I should use the boundness of the sequence... but I cannot see how it will work.
Suggestions?