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MATH 301 Exercise 11.2

• 1. INFINITE SERIES CHAPTER 11
• 2. Convergent or Divergent Series Exercise 11.2 Page 541
• 3. IN THIS LESSON YOU WILL LEARN:  Difference between a sequence and a series  Notation of a sequence and a series  Infinite geometric series  Convergent series  Divergent series  Sum of convergent series  Nonterminating decimal (Recurring Decimal)
• 4. WHAT’S THE DIFFERENCE BETWEEN A SEQUENCE AND A SERIES? A sequence is a list (separated by commas). A series adds the numbers in the list together. Example: Sequence: 1, 2, 3, 4, …, n, … Series: 1 + 2 + 3 + 4 + …+ n + … (Note that in calculus we only examine infinite sequences and series)
• 5. WHAT SYMBOL(S) DO WE USE For a sequence? { an } represents a sequence. For a series? ∑ an represents a series.
• 6. An INFINITE SERIES (or simply a series) is an expression of the form a1 + a 2 + a 3 + ..... + an + ...
• 7. RULE Let a ≠ 0 . The geometric series a + ar + ar 2 + .... + ar n− 1 + ..... (i) Converges and has the sum a S= , if r < 1 1− r (ii) Diverges if r ≥1
• 8. EXAMPLE 5 PAGE 537 Prove that the following series converges , and find its sum: 2 2 2 2 + + 2 + .... + n−1 + .... 3 3 3
• 9. SOLUTION The series converges , since it is geometric with r < 1. Here 1 r = <1 a = 2 and 3 . The sum is a 2 2 s= = = =3 1− r 1− 1 2 3 3
• 10. OR RECURRING DECIMALS All nonterminating (recurring) decimals can be written as fractions They are ‘rational numbers’
• 11. RECURRING DECIMALS Recurring decimals are written by using a dot: • 0.33333 ....... = 0. 3 • • 0.35353535 ....... = 0. 3 5
• 12. EXERCISE 11.2 PAGE 541 Question12 Determine whether the geometric series converges or diverges; if it converges, find its sum. 628 0.628 + 0.000628 + .... + n + ... (1000 )
• 13. SOLUTION The series converges , since it is geometric with r < 1. Here 1 r = <1 a = 0.628 and 1000 . The sum is a 0.628 628 s= = = 1−r 1 999 1− 1000
• 14. EXERCISE 11.2 PAGE 541 Question 55 A rubber ball is dropped from a height of 10 meters. If it rebounds approximately one-half the distance after each fall, use a geometric series to approximate the total distance the ball travels before coming to rest.
• 15. SOLUTION  5 5  S = 10 + 2 5 + + + .......  2 4   1 1  S = 10 + 2 × 5 1 + + + .......  2 4    10m 5m  1  5 m ….. S = 10 + 10  2 1− 1     2 S = 10 + 20 = 30 Meters
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