Hey there,

I'll be glad to get some help in the following questions:

1. Let g(x) be a differentiable function at $\displaystyle (0,1]$ which satisfies: $\displaystyle x^{2}g(x) $ is a monotonic ascending function at

$\displaystyle (0,1]$ and: $\displaystyle lim_{x \to 0^{+}} x^{2}g(x)=0 $, g'(x) is continous at $\displaystyle (0,1]$.

Check whether the integral $\displaystyle \int_{0}^{1} g(x)sin(\frac{1}{x})dx $ converges.

2. Let f(x) be a function defined by: $\displaystyle f(x)=\Sigma_{n=1}{\infty}a_{n}sin(nx) $ .

Prove that if the series $\displaystyle \Sigma_{n=1}^{\infty}n|a_{n}| $ converges then f(x) is differentiable for every real x.

Thanks in advance