Exercise Show that the series $\displaystyle\sum_{n=1}^\infty \frac{x^2}{x^3+n^3}$ is uniformly convergent on $[0,\infty)$.
More precisely we define $f_n(x) : = \frac{x^2}{x^3+n^3}$ for $n \in \mathbb{N}$ and we we seek to show that there exists some function $f(x)$ such that the sequence of functions $\langle f_n \rangle$ converge to $f$ uniformly on $[0,\infty)$.
Any hints or tips on how to approach this? It is a well known fact that power series
$$\displaystyle\sum_{k=1}^\infty a_k(x-c)^k$$
converge uniformly and absolutely on their interval of convergence $(c-R,c+R)$. Any advice?
This seems false: let $R_n(x)=\sum_{p > n}{\frac{x^2}{x^3+p^3}}$. Then $$ R_n(n) \geq \sum_{p=n+1}^{2n}{\frac{n^2}{n^3+p^3}} \geq \sum_{p=n+1}^{2n}{\frac{n^2}{9n^3}} \geq 0.1 $$ So $R_n$ cannot converge uniformly to zero.