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A geometric series is a series whose related sequence is geometric. It results from adding the terms of a geometric sequence .
Example 1:
Finite geometric sequence:
Related finite geometric series:
Written in sigma notation:
Example 2:
Infinite geometric sequence:
Related infinite geometric series:
Written in sigma notation:
To find the sum of a finite geometric series, use the formula,
,
where
is the number of terms,
is the first term and
is the
common ratio
.
Example 3:
Find the sum of the first terms of the geometric series if and .
Example 4:
Find , the tenth partial sum of the infinite geometric series .
First, find .
Now, find the sum:
Example 5:
Evaluate.
(You are finding for the series , whose common ratio is .)
To find the sum of an infinite geometric series having ratios with an
absolute value
less than one, use the formula,
,
where
is the first term and
is the common ratio.
Example 6:
Find the sum of the infinite geometric series
.
First find :
Then find the sum:
Example 7:
Find the sum of the infinite geometric series
if it exists.
First find :
Since is not less than one, the series does not converge. That is, it has no sum.