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1/31/08 8:56 AM
The Geometric Series
The easiest (but not the only) way is to factor out
:
The series inside the parentheses is the familiar geometric series with
. Thus, this series sums to
For which values of
q
will this work?
The summation trick on the previous page does
not
work for all values of
q
. Consider for instance
q
=1. Clearly,
the sum
does not add up to a finite number! One says that this series
diverges
(= is not convergent). This does not have
much to do with the fact that in the end we "divide by 0"; try
q
=2 or
q
=1.
The problem lies much deeper. The sad truth is that many of the algebraic properties of finite sums do not work
for infinite sumstroubling mathematicians over the centuries! So let's be very cautious and try again. This time
we only consider finite sums and then take the limit! Let
multiply both sides by
q
then subtract the second line from the first:
For
, we can solve this for
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 Winter '08
 JaberAbdualrahman
 Addition, Geometric Series, finite sums

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