If $\,p$ and $\,q$ are real numbers,the following series is convergent for.... I am stuck on the following problem :

If $p$ and $q$ are positive real numbers, the series $$\frac{2^p}{1^q}+\frac{3^p}{2^q}+\frac{4^p}{3^q}+\ldots$$ is convergent for
$\quad(a)\quad p<q-1$
$\quad(b)\quad p<q+1$
$\quad(c)\quad p\ge q-1$
$\quad(d)\quad p\ge q+1$

Can someone explain how to tackle it? Thanks and regards to all.

The series can be written as $\sum_{n=1}^{+\infty}\frac{(n+1)^p}{n^q}$.

The way to solve this problem is to compare it, using the limit comparison test, with a series of the form $\sum_{n=1}^{+\infty}\frac{1}{n^r}$ for suitable $r$.
