I would like to find tight upper and lower bounds for the sum $S(x)$ of the following series of functions:
$$ S(x)=\sum_{n=1}^\infty f_n(x)=\sum_{n=1}^\infty\frac{c_n}{(x+1)^n},\qquad x\geq0, $$
where $c_n$ is a bounded sequence of real numbers, $0\leq c_n\leq1$ for every $n\geq1$, and asymptotically $c_n\sim n^{-\frac{d}{2}}$, and $d$ is a positive integer.
Notice that, if $d\geq3$, $\sum_{n=1}^\infty f_n(x)$ converges poitwise on the whole $[0,+\infty[$, and hence $S(x)$ is well defined on the whole $[0,+\infty[$ for these values of $d$; otherwise ($d=1,2$), $\sum_{n=1}^\infty f_n(x)$ converges only on $]0,+\infty[$.
Moreover, notice that, if $d\geq3$, the just mentioned pointwise convergence is total (hence uniform) on $[0,+\infty[$, while if $d=1,2$, it is total just on every $[\epsilon,+\infty[$, $\epsilon>0$.
Any suggestions on how I could proceed?