In a solution of a book of the integral:
$$\int_a^{\infty} \sum_{n=1}^{\infty} \frac{1}{(z+n)^{k+1}}\,dz, \;\; a\geq 1$$
I see the following:
$$\begin{align*} \int_{a}^{\infty}\sum_{n=1}^{\infty}\frac{1}{\left ( n+z \right )^{k+1}}\,dz &= \sum_{n=1}^{\infty}\int_{a}^{\infty}\frac{dz}{(n+z)^{k+1}}\\ &= \cdots\\ \end{align*}$$
The rest of the solution is understable to me but not the interchange. I was unable to prove that the fuction within the series converges uniformly... and I cannot think of something else that works here e.g monotone convergance thoerem or Tonelli Theorem.
After the discussion above the answer to my question is:
We are allowed to interchange summation and integration by the use of Tonelli's Theorem because that thing that lies in the series is positive.
And everything is going its way.
Thanks for the help!!