convergence of $\frac{x^n}{1+x^n}$

Question

How do I check convergence/ uniform convergence of $\sum\frac{x^n}{1+x^n}$.

Also for series $\sum \sin \left(\frac{x}{n^2}\right)$, can I use that $\sin x \leq x$?

Answer 1

For the first series we have a point-wise convergence on the interval $(-1,1)$ since for $-1<x<1$ we have $$\frac{x^n}{1+x^n}\sim_\infty x^n$$ and the geometric series $\sum_n x^n$ is convergent. There's not a unifrom convergence on this interval because the limit $$\lim_{x\pm1}\sum_{n=1}^\infty \frac{x^n}{1+x^n}$$ doesn't exist.

For the second series we have $$\sin\left(\frac x{n^2}\right)\sim_\infty \frac x{n^2}$$ and the Riemann series is convergent. There's not a uniform convergence on $\Bbb R$ because the series $$\sum_{n=1}^\infty\sin\left(\frac1n\right)$$ is divergent.