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.