9­5 Mathematical Induction and Infinite Geometric Series­ No Induction­ period 1.notebook
April 29, 2015
Bellwork 4-28-15
Evaluate.
1.
2.
3. Write 0.6 as a fraction in simplest form.
4. Find the indicated sum for the geometric series
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9­5 Mathematical Induction and Infinite Geometric Series­ No Induction­ period 1.notebook
April 29, 2015
Objectives
Find sums of infinite geometric series.
Use mathematical induction to prove statements.
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9­5 Mathematical Induction and Infinite Geometric Series­ No Induction­ period 1.notebook
April 29, 2015
Vocabulary
infinite geometric series
converge
limit
diverge
mathematical induction
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9­5 Mathematical Induction and Infinite Geometric Series­ No Induction­ period 1.notebook
April 29, 2015
An infinite geometric series has infinitely many terms. Consider the
two infinite geometric series below.
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9­5 Mathematical Induction and Infinite Geometric Series­ No Induction­ period 1.notebook
April 29, 2015
Notice that the series Sn has a common ratio of and the partial
sums get closer and closer to 1 as n increases. When |r|< 1 and the
partial sum approaches a fixed number, the series is said to converge.
The number that the partial sums approach, as n increases, is called a
limit.
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9­5 Mathematical Induction and Infinite Geometric Series­ No Induction­ period 1.notebook
April 29, 2015
For the series Rn, the opposite applies. Its common ratio is 2, and its
partial sums increase toward infinity. When |r| ≥ 1 and the partial
sum does not approach a fixed number, the series is said to diverge.
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9­5 Mathematical Induction and Infinite Geometric Series­ No Induction­ period 1.notebook
April 29, 2015
Example 1: Finding Convergent or Divergent Series
Determine whether each geometric series converges or diverges.
A. 10 + 1 + 0.1 + 0.01 + ...
B. 4 + 12 + 36 + 108 + ...
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9­5 Mathematical Induction and Infinite Geometric Series­ No Induction­ period 1.notebook
April 29, 2015
Check It Out! Example 1
Determine whether each geometric series converges or diverges.
A.
B. 32 + 16 + 8 + 4 + 2 + …
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9­5 Mathematical Induction and Infinite Geometric Series­ No Induction­ period 1.notebook
April 29, 2015
9
9­5 Mathematical Induction and Infinite Geometric Series­ No Induction­ period 1.notebook
April 29, 2015
Example 2A: Find the Sums of Infinite Geometric Series
Find the sum of the infinite geometric series, if it exists.
1 – 0.2 + 0.04 – 0.008 + ...
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9­5 Mathematical Induction and Infinite Geometric Series­ No Induction­ period 1.notebook
April 29, 2015
Example 2B: Find the Sums of Infinite Geometric Series
Find the sum of the infinite geometric series, if it exists.
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9­5 Mathematical Induction and Infinite Geometric Series­ No Induction­ period 1.notebook
April 29, 2015
Check It Out! Example 2a
Find the sum of the infinite geometric series, if it exists.
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9­5 Mathematical Induction and Infinite Geometric Series­ No Induction­ period 1.notebook
April 29, 2015
Find the sum of the infinite geometric series, if it exists.
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9­5 Mathematical Induction and Infinite Geometric Series­ No Induction­ period 1.notebook
April 29, 2015
Homework: Worksheet 9-5 Practice B, #'s 1-6
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