4
votes
1answer
146 views

Estimate the scale of $e^{-(m+1) t} \sum _{k=0}^{\infty } \frac{t^k}{k!}\left(\sum _{r=0}^k \frac{t^r}{r!}\right)^{m}$

I would like to estimate the scale of the following series, $$S(m,t)=e^{-(m+1) t} \sum _{k=0}^{\infty } \frac{t^k}{k!}\left(\sum _{r=0}^k \frac{t^r}{r!}\right)^{m},$$ where $e$ is the base of ...
3
votes
1answer
173 views

Solving perturbed polynomial equations

Rather than asking the most general question possible, I will frame it in terms of what I believe is an illustrative example. Let $\epsilon>0$ be a small parameter, let $a,b>0$ and $x\in ...