11­5 Geometric Series
April 28, 2009
11‑5 Geometric Series
Objective: Write and evaluate geometric series.
Apr 21­4:15 PM
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11­5 Geometric Series
April 28, 2009
Check Skills You'll Need
Find each sum or difference.
1. 100 + 50 + 25 + 25/2 + 25/4
3. ­2 + 4 ­ 8 + 16 ­ 32
2. 3 + 9 + 27 + 81
4. ­5 ­ 10 ­ 20 ­ 40
Simplify each fraction.
5.
1 ­ 1/5
1/3
6.
1
1 ­ 1/4
7. 1/2 ­ 1/3
1/4
Apr 26­5:25 PM
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11­5 Geometric Series
April 28, 2009
A Geometric Series is the expression for the sum of the terms of a geometric sequence.
There are two types of geometric series:
Finite (it stops)
Infinite (has no end)
2 + 4 + 8 + 16
2 + 4 + 8 + 16 ...
Apr 26­5:34 PM
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11­5 Geometric Series
April 28, 2009
As with arithmetic series, you can use a formula to evaluate a finite geometric series.
Apr 26­5:45 PM
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11­5 Geometric Series
April 28, 2009
Example #1: Using the Geometric Series Formula
Use the formula to evaluate the series 3 + 6 + 12 + 24 + 48 + 96.
What is the first term (a1)?
How many terms are there (n)?
What is the common ratio (r)?
Plug the numbers into the formula: a1(1 ­ rn) Sn = 1 ­ r Apr 26­5:32 PM
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11­5 Geometric Series
April 28, 2009
Evaluate each series.
a. ­45 + 135 ­ 405 + 1215 ­ 3645
a1(1 ­ rn) Sn = 1 ­ r b. 1/3 + 1/9 + 1/27 + 1/81
Apr 26­5:43 PM
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11­5 Geometric Series
April 28, 2009
In some cases, you can evaluate an infinite geometric series. When |r| < 1, the series converges, or gets closer and closer to the sum, S.
When |r| ≥ 1, the series diverges, or approaches no limit.
Apr 26­5:50 PM
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11­5 Geometric Series
April 28, 2009
Example #2 Determining Divergence and Convergence
Decide whether each infinite geometric series diverges or converges. State whether the series has a sum.
a. 1 ­ 1/3 + 1/9 ­...
b. ∞
5(2)n­1
n = 1
Since |r| < 1, the series
converges, and the series has a sum. Since |r| ≥ 1, the series
diverges, and the series does not have a sum. Apr 26­5:55 PM
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11­5 Geometric Series
April 28, 2009
As with a finite geometric series, you can use a formula to evaluate an infinite geometric series if |r| < 1.
Apr 26­6:02 PM
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11­5 Geometric Series
April 28, 2009
Example #3 Using the Infinite Geometric Series Formula
Evaluate the infinite geometric series 1 + 1/2 + 1/4 + 1/8...
What is the first term (a1)?
What is the common ratio (r)?
Plug the numbers into the formula: S = a1 1 ­ r Apr 26­6:05 PM
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11­5 Geometric Series
April 28, 2009
Recap:
What is the formula for the sum of a FINITE geometric series?
a1(1 ­ rn) Sn = 1 ­ r What is the formula for the sum of an INFINITE geometric series that converges?
a1 S = 1 ­ r Why is there no formula for an INFINITE geometric series that diverges?
Apr 26­6:11 PM
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11­5 Geometric Series
April 28, 2009
Homework
p. 628 #2­28 even, 32­37
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