Calculate the following limit ( not sure if possible)

[csopi]

Homework Statement

Calculate the following limit for real $t$-s.

$$\sum_{n=0}^{∞} exp[i\cdot \sqrt{n + 1}\cdot t] / n!$$

Homework Equations

None

The Attempt at a Solution

Without the root it's trivial... I am not sure if it is even possible to give a closed form, I am out of ideas. Any help would be greatly appreciated!

 

[jbunniii]

What have you tried? A reasonable first step would be to put $m = \sqrt{n+1}$. What does that give you?

 

[csopi]

Thank you for your efforts! I've tried that, but I think it won't help, because m won't be an integer. I also tried to approximate with an intergral using Stirling's formula for n!, but the resulting intergral seems too complicated. I'm also considering to use somehow the residue theorem, but so far nothing.

 

[StevePMcGrew]

You can build an Excel spreadsheet in about five minutes to do the calculation to a very good approximation:
For T=pi, for example, the limit is approximately Lim = .539061035756653 -i*.335197295005148.

 

[haruspex]

Not sure whether it provides any useful clues, but here are a couple of plots. One plots y against x (i.e. complex plane), the other plots r and theta as functions of t. For the second, I normalised r by dividing by the t=0 value (e), and normalised theta by subtracting t (which seems to be the asymptotic behaviour) then dividing by pi.