Bounding integral of the form $\int_0^\infty |z+x|^{-n}dx$ I need to bound a sum of the form $\sum_{j=1}^m |z+j|^{-n}$ with $\Re z\ge1$ and $n \ge 2$. I am searching for a bound of the form 
$$\sum_{j=1}^m |z+j|^{-n} \le C |z|^{-n+1}.$$
It is easy to see that
$$\sum_{j=1}^m |z+j|^{-n} \le \int_{0}^{m+1} |z+x|^{-n}dx \le \int_{0}^{\infty} |z+x|^{-n}dx.$$
My problem is that I cannot compute the integral. By setting $z=r+i s$, we have
$$\int_0^\infty ((r+x)^2+s^2)^{-\frac n2} dx.$$
My idea was to use integration by parts to transform it to an integrable form, but so far I cannot find how I am going to do that. Any help is welcome.
 A: We have $(r+x)^2 + s^2 \geq (r+x)^2$. Hence, we have
$$\int_0^{\infty} \dfrac{dx}{((r+x)^2 + s^2)^{n/2}} \leq \int_0^{\infty} \dfrac{dx}{(r+x)^n} = \dfrac{r^{-n+1}}{n-1} < r^{-n+1}$$
We also have $(r+x)^2 + s^2 \geq x^2+s^2$. Hence, we have
\begin{align}
\int_0^{\infty} \dfrac{dx}{((r+x)^2 + s^2)^{n/2}} & \leq \int_0^{\infty} \dfrac{dx}{(x^2+s^2)^{n/2}} = \int_0^{\pi/2} \dfrac{\vert s \vert \sec^2(t) dt}{\vert s \vert^n \sec^n(t)}\\
& = \vert s \vert^{-n+1} \int_0^{\pi/2} \cos^{n-2}(t)dt < \vert s \vert^{-n+1} \dfrac{\pi}2
\end{align}
Hence, we have (make use of the fact that $\vert r \vert + \vert s \vert \leq 2 \vert z\vert$)
$$\int_0^{\infty} \dfrac{dx}{((r+x)^2 + s^2)^{n/2}} \leq C \vert z \vert^{-n+1}$$
A: $\newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
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$\large\it\mbox{Hint:}$

$\large n > 1:$

\begin{align}
{\cal T}_{n}
&=
\int_{0}^{\infty}{\dd x \over \bracks{\pars{r + x}^2 + s^{2}}^{n/2}}
=
{n \over 2}\int_{0}^{\infty}
{x\bracks{2\pars{r + x}} \over \bracks{\pars{r + x}^2 + s^{2}}^{n/2 + 1}}\,\dd x
\\[3mm]&=
n\int_{0}^{\infty}
{\bracks{\pars{r + x}^{2} + s^{2}} - \bracks{r\pars{r + x} + s^{2}} \over \bracks{\pars{r + x}^2 + s^{2}}^{n/2 + 1}}\,\dd x
\\[3mm]&=
n\int_{0}^{\infty}\!\!\!\!\!\!
{\dd x \over \bracks{\pars{r + x}^2 + s^{2}}^{n/2}}\,
+
\left.r\,
{1 \over \bracks{\pars{r + x}^2 + s^{2}}^{n/2}}\right\vert_{0}^{\infty}
-
ns^{2}\int_{0}^{\infty}\!\!\!\!\!
{\dd x \over \bracks{\pars{r + x}^2 + s^{2}}^{n/2 + 1}}\,
\\[3mm]&=
n{\cal T}{n}
-
{r \over \bracks{r^2 + s^{2}}^{n/2}}
-
ns^{2}{\cal T}_{n + 2}
\end{align}


\begin{align}
{\cal T}_{n + 2}
&\equiv
{1 \over s^{2}}\,{n - 1 \over n}\,{\cal T}_{n}
-
{r \over n s^{2}}\,{1 \over \pars{r^{2} + s^{2}}^{n/2}}
\\[3mm]
{\cal T}_{2}
&=
\int_{0}^{\infty}{\dd x \over \pars{r + x}^{2} + s^{2}}
=
{1 \over \verts{s}}\,\bracks{{\pi \over 2} - \arctan\pars{r \over \verts{s}}}
\\[3mm]
{\cal T}_{3}
&=
\int_{0}^{\infty}{\dd x \over \bracks{\pars{r + x}^{2} + s^{2}}^{3/2}}
=
{1 \over s^{2}}\,\pars{1 - {r \over \sqrt{r^{2} + s^{2}\,}\,}}
\end{align}

