Alright, sorry about this, I was able to solve it. I wish I could delete the thread, however here's the solution just in case anyone is wondering I guess.
We know . Here and . Thus as desired.
Problem: Using the Taylor series for , and assuming that the Laplace transform of this series can be computed term by term, verify that
Attempt at solution:
So I start with the the taylor series of ,
Then using the fact that where , I am able to get to this step:
However here I'm stumped, I'm guessing there's some well known summation result I should be using to get to the final step, but the ones I've looked up I haven't been able to successfully use them to get the result I want.
Any suggestions would greatly be appreciated. Thanks in advance.