uniform convergence of $S_n=\frac{nx}{1+n^2x^3}$

Question

comment about the uniform convergence of the series, whose partial sum of n terms is $S_n=\frac{nx}{1+n^2x^3}$ , for all real x.

for showing uniform convergence, I tried to use the definition $∀\epsilon>0$: $∃ m∈N$ such that $|f_n(x)−f(x)|<ϵ \ \ \forall n>m(ϵ)$.

but unable to get any expression in terms of n for which I can find the supremum and show uniform convergence, also how should I work with an interval so that I can get a feasible solution?

any hint would be highly appreciated.

ps: so its not uniform convergent for R THEN what interval should be taken such that it is uniform convergent

Answer 1

Take $x_n = \frac1{n^{\frac23}}$ can you see that the limit of $S_n(x_n)$ is $\infty$?

Answer 2

You can show $$ k_n=\sup_{x \in \mathbb{R}} \bigg| \frac{nx}{1+n^2x^3}\bigg|=\infty$$ This follows it is not U.C.