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

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

Thx in advance...

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- Mar 31st 2009, 02:20 PMwheatkcseries 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).**

Thx in advance...

- Apr 1st 2009, 05:19 AMchisigma
The Weiestrass M-test can be used in this case. If $\displaystyle x \in [a,b]$ and we set $\displaystyle c= max [|a|,|b|]$ is…

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

But the series…

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

… is absolutely convergent so that the series…

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

… is uniformly convergent in $\displaystyle [a,b]$…

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$