# Proving whether function-series $f_n(x) = \frac{(-1)^nx}n$

I'm asked to prove whether the following function-series on $E$ with values in $Y$ converges pointwise/uniformly/absolutely.

$$E = [-1, 1], Y= \Bbb{R}$$

$$f_n = \frac{(-1)^nx}n.$$

What I have tried so far: Using direct comparison test to find a convergent series and then using the Weierstrass M-Test but I can't find any solution to this problem.

• you dont find any solution to any of the asked questions? Observe that $$\sum_{k=1}^\infty\frac{(-1)^k x}k=x\sum_{k=1}^\infty\frac{(-1)^k}k$$ What about it pointwise convergence? And it absolut convergence? – Masacroso Jan 13 '18 at 14:33
• @Masacroso Good idea. Does that imply that $f_n$ is not converging absolutely (harmonic series) but convering uniformly and therefore pointwise? – m4rkus Jan 13 '18 at 14:41
• Yes that's it ! – Atmos Jan 13 '18 at 14:45