$ \sum_{n=1}^\infty \frac{x^4}{(1+x^4)^n} $ converges pointwise on $ R$, but not uniformly on $R$.

Question

Prove Or Disprove, The series summation $$ \sum_{n=1}^\infty \frac{x^4}{(1+x^4)^n} $$ converges pointwise on $ R$, but not uniformly on $R$.

I am struggling on this problem in real analysis analysis. I know some how I have use weirstrass M test. but could not see it clearly. How to get start ?

Answer 1

(I don't think you have to use Weierstrass M-test, since one cannot conclude that a series is not uniformly convergent from that test)

Hint: Calculate the series, both for $x\neq 0$ and $x=0$. Do you have any knowledge of what kind of function a series of continuous functions that converge uniformly must converge to?

Update If you have a series of functions where each function is continuous and the series is uniformly convergent, then the function the series converges to must be continuous.

Note that your series is $0$ for $x=0$ since each term is zero. If $x\neq 0$, you have a geometric series. Calculate it!