Thread: 2 calculus quesions

1. 2 calculus quesions

Hey there,

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

1. Let g(x) be a differentiable function at $(0,1]$ which satisfies: $x^{2}g(x)$ is a monotonic ascending function at
$(0,1]$ and: $lim_{x \to 0^{+}} x^{2}g(x)=0$, g'(x) is continous at $(0,1]$.
Check whether the integral $\int_{0}^{1} g(x)sin(\frac{1}{x})dx$ converges.

2. Let f(x) be a function defined by: $f(x)=\Sigma_{n=1}{\infty}a_{n}sin(nx)$ .
Prove that if the series $\Sigma_{n=1}^{\infty}n|a_{n}|$ converges then f(x) is differentiable for every real x.

Thanks in advance

2. Setting $\frac{1}{x}=t$ the integral becomes...

$\int_{0}^{1} g(x)\cdot \sin \frac{1}{x}\cdot dx = \int_{1}^{\infty} g(\frac{1}{t})\cdot \frac{\sin t}{t^{2}}\cdot dt$ (1)

Now the function...

$\gamma (t) = \frac{g(\frac{1}{t})}{t^{2}}$ (2)

... is monotonically decreasing for $t>1$ nd ...

$\lim_{t \rightarrow \infty} \frac{g(\frac{1}{t})}{t^{2}}=0$ (3)

... so that is...

$|\int_{1}^{\infty} g(\frac{1}{t})\cdot \frac{\sin t}{t^{2}}\cdot dt| < |\int_{1}^{\pi} \gamma(t)\cdot \sin t\cdot dt| + \pi \cdot |\sum_{n=1}^{\infty} (-1)^{n} \gamma (n\cdot \pi)|$ (4)

... and the integral converges...

Kind regards

$\chi$ $\sigma$

3. Thanks a lot