Help showing bound for magnitude of complex log fcn

[benorin]

I'm working through an example problem wherein this bound is used:

\left| \log \left( 1-\frac{1}{L^s}\right) \right| \leq L^{-\sigma},

where s:=\sigma +it and it is known that \sigma >1. How do I prove this? Should I assume the principle brach is taken?

 

[benorin]

It is perhaps important to note that in the context, L\rightarrow\infty

I think I got a start:

For \left| z\right| <1, the Taylor series about z=0 is

\log (1-z)=-\sum_{j=1}^{\infty}\frac{z^j}{j}

so that

\left| \log \left( 1-\frac{1}{L^s}\right) \right| =\left| -\sum_{j=1}^{\infty}\frac{L^{-js}}{j} \right| \leq \sum_{j=1}^{\infty}\left| \frac{L^{-js}}{j}\right| = \sum_{j=1}^{\infty} \frac{L^{-j\sigma}}{j}