Monotonically Decreasing Sequence

Question

Let $$b_n=|\sum_{m=1}^L \frac{a_m}{(n+1)^{x_m}}-\frac{a_m}{n^{x_m}}| $$ with $L<\infty$, $a_m,x_m \in \Bbb C$ and $Re(x_m)>0$

I would like to know if $b_{n+1}<b_n$

Thanks.

Answer 1

No, it's not even true for $L=1$. Take $a_m=1$ and $n=1$ and $x_m = 1+2\pi i/\log 2$.