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.
$1)$ I know that the improper integral $ \int_a^\infty f(x) \, dx$ is convergent if and only if the function $F(b) = $$ \int_a^b f(x)\,dx$ is bounded. I have found the indefinite integral of $$ \int \frac {\ln(x)}{x^p} = \frac {x^{1-p} \ln(x)}{1-p}- \frac {x^{-p+1}}{(1-p)^2} + C $$
How do I prove that is bounded and determine its value?
$2)$ I know that I can make use of the integral criteria: The infinite series $\sum_{n=1}^\infty = f(n)$ converges if and only if the improper integral $ \int_1^\infty f(x)\,dx$ is convergent.
I am uncertain as to how I can show this.