Let a be any real number. Then the series
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is called Geometric Series.
- if | a | < 1 the geometric series converges
- if | a |
1 the geometric
series diverges
If the geometric series converges (i.e. if | a | < 1) then
- =
Note that the index for the geometric series starts at 0. This is not important for the
convergence behavior, but it is important for the resulting limit.
Examples:
Investigate the convergence behaviour of the following series:
- What is the actual limit of the sum
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?
- What is the actual limit of the sum
?
- Does the sum
converge ?
(Here the limit comparison test may be helpful).
Proof:
The proof consists of a nice trick. Consider the partial sum
S N and multiply it by a:
- S N =
1 + a + a 2 + a 3 + ... + a N
- a S N =
a + a 2 + a 3 + ... + a N+1
Subtracting both equations yields:
(1 - a) SN = 1 - a N+1.
Dividing both sides by (1 - a) and taking the limit the result
follows from previous result on the
power sequence.
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