Laplace Transform Determination/Series Method

Solution Technique for Laplace Transform
To find the Laplace transform of a function $f$, one can evaluate it as follows:

Let $\map f t$ have a power series expansion given by:

Then the Laplace transform of $f$ can be found by taking the sum of the Laplace transforms of each term in the series:

if $\laptrans {\map f t}$ is convergent for $s > \gamma$.

Proof
From Laplace Transform of Positive Integer Power:
 * $\laptrans {t^n} = \dfrac {n!} {s^{n + 1} }$

Then it is seen that $\ds \laptrans {\sum_{n \mathop = 0}^\infty a_n t^n}$ is a Linear Combination of Laplace Transforms.

The result follows.