Evaluate the following infinite sum with parameter So the sum actually originates from the following integral:
$$\int_0^\infty \frac{\left\lfloor x \right\rfloor}{x^{a + 1}}dx$$
where $a$ is a real parameter. I've managed to transform the integral into the following sum (except for the case $a = 0$, which is observed separately), which actually became a lot more interesting for me:
$$\sum_{n=0}^\infty \frac{n}{a}(\frac{(n+1)^a - n^a}{n^a(n+1)^a})$$
But I'm clueless as to how to approach that sum? Any ideas? I'm really keen on figuring this out! Many thanks in advance!

$HINT$
$n\frac{(n+1)^a-n^a}{n^a(n+1)^a}=\frac{n}{n^a}-\frac{n+1-1}{(n+1)^a}=\frac{1}{n^{a-1}}-\frac{1}{(n+1)^{a-1}}+\frac{1}{(n+1)^a}$
Now take cases for $a$
For $a=1$ the series diverge.
