I am trying to solve find a solution to a). i can use maple to solve it, but does anyone know how. The "sum of" part is confusing me. Thanks in advance for any help.
You should be familiar with the Dirac delta function and its properties, in particular the sifting property: .
You should also be familiar withy the definition of a Laplace transform: .
Using the above you should be able to show that .
Alternatively you can always refer to a standard table of Laplace transforms.
Therefore the Laplace transform of your function is .
Use the formula for the sum of a geometric series to evaluate this sum.