Geometric Series

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A geometric series is formed by multiplying each term by a constant.

For example

 each term is doubled
This series can be broken down to
So that each term can be written as
With a as the starting term and r as the multiplying factor.

r is called the common ratio.

A second example.

Write the first 4 terms of the geometric series described by
Substitute into

Finding the Sum of a Geometric Series
The following, ingenious method is used to find the sum of a geometric series
Write out Sn
Multiply by r. Notice how most terms pair up.
Subtract the two equations.
Factorize the left hand side.
Divide by r-1
Either arrangement will work, but the second is more appropriate if r<1

Example: Find the sum of the first 6 terms of the series
Define a, r and n
Substitute into formula.

The Sum To Infinity
In the case of
then
Our sum formula then becomes
For example: Find the sum of the first 8 terms of the following series and the sum to infinity
Define a, r, n
Substitute into formula
For the sum to infinity
Note that
so the series was not far from the sum to infinity after 8 terms.

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