Problem: Using the Taylor series for $\displaystyle \sin t$, and assuming that the Laplace transform of this series can be computed term by term, verify that $\displaystyle \mathcal{L}\{\sin t\} = \frac{1}{s^2 + 1}.$

Attempt at solution:

So I start with the the taylor series of $\displaystyle \sin t = \displaystyle\sum^{\infty}_{n=0} \frac{(-1)^n t^{2n + 1}}{(2n + 1)!}$,

Then using the fact that $\displaystyle \mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}},$ where $\displaystyle n \in \mathbb{Z}^+$, I am able to get to this step:

$\displaystyle \mathcal{L}\{\sin t\} = F(s) = \displaystyle\sum^{\infty}_{n=0} \frac{(-1)^n}{s^{2(n+1)}}$

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.