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MATH 301 Exercise 11.2

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INFINITE SERIES
  CHAPTER 11
Convergent or Divergent
Series
       Exercise 11.2
         Page 541
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)
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)
WHAT SYMBOL(S) DO WE
           USE

For a sequence?


 { an }     represents a sequence.



For a series?


  ∑ an          represents a series.
An   INFINITE       SERIES
(or simply a series) is an
expression of the form
 a1 + a 2 + a 3 + ..... + an + ...

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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