## 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.

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