I want to calculate characteristic function of following discrete distribution: $$P(X=m)=\frac{a^{m}}{(a+1)^{m+1}}$$
where $m=0,1,2......$ and $a>0$.
$$\varphi _{X}(t)=\frac{e^{it}}{(a+1)}\sum_{m=0}^{\infty}\frac{(ae)^{m}}{(a+1)^{m}}$$
I tried to reduce it to sum of geometric series, but I am stuck at this point.
Taking Did's comment, let's demonstrate that $\left|\dfrac {ae^{it}}{a+1}\right|\lt1$.
$|ae^{it}|=|a|\cdot |e^{it}|=|a|\cdot|\cos(t)+i\sin(t)|=|a|\sqrt{\cos^2(t)+\sin^2(t)}=|a|$
Thus $\left|\dfrac {ae^{it}}{a+1}\right|=\left|\dfrac a{a+1}\right|=\dfrac a{a+1}\lt1$