1. ## series convegence

Show that (sum from 1 to infinity) sin(x/n^2) converges uniformly on any closed interval [a, b].

I think I can use Weierstrass M-Test to prove this, but I can't find the limit for fn(x).

2. The Weiestrass M-test can be used in this case. If $x \in [a,b]$ and we set $c= max [|a|,|b|]$ is…

$|\sin (\frac{x}{n^{2}})|\le \frac{c}{n^{2}}$

But the series…

$\sum_{n=1}^{\infty} \frac {c}{n^{2}}$

… is absolutely convergent so that the series…

$\sum_{n=1}^{\infty} \sin (\frac{x}{n^{2}})$

… is uniformly convergent in $[a,b]$

Kind regards

$\chi$ $\sigma$