Uniform Convergence using Abel's test for a series based on convergence of a series

A problem from uniform convergence of series:$$\sum_{i=1}^\infty a_n$$ is convergent then show that $$\sum_{i=1}^\infty \frac {nx^n(1-x)}{1+x^n} a_n$$ and $$\sum_{i=1}^\infty \frac {2nx^n(1-x)}{1+x^{2n}} a_n$$ are uniformly convergent when $x \in\ [0,1]$.