# Convergence of improper integral and infinite series

As $p>1$ is a real number, the function $f$ is defined as $$f(x)= \frac {\ln(x)}{x^p}\,,x>0$$

$1)$Show that the improper integral $$\int_a^\infty \frac {\ln(x)}{x^p} \, dx$$ is convergent for $a>0$, and determine its value.

$2)$ Show that the infinite series

$$\sum_{n=1}^\infty \frac{\ln(n)}{n^p}$$ is convergent.