The Geometric Series

Since we use the geometric series constantly when dealing with Laplace
transforms, it's important to be very familiar with it.  In general
a geometric series is of the form

       S  =  1 + x + x^2 + x^3 + x^4 + ...

Assuming this infinite series converges to a finite value (which it 
does for any x less than 1) we can easily derive a closed-form 
expression for this sum.  Notice that S can be written in the form

       S  =  1  +  x (1 + x + x^2 + x^3 + ...)

and the quantity in parentheses is S.  Thus we have S = 1 + xS,
and so
                       1
                S =  -----
                     1 - x

Incidentally, this little trick was known to Euclid (circa 300 BC), 
and can be found in his treatment of "perfect numbers" in Book 9 of 
"The Elements".  In a sense, Laplace transforms are the same basic
"trick", but applied to derivatives instead of powers, as discussed 
in the main article on Laplace transforms.