1. AP Calculus March 22, 2010 Berkley High School, D1B1
Section 12.2 Series
AP Calculus
March 22, 2010
Berkley High School, D1B1

2. Calculus, Section 12.2, Todd Fadoir, CASA, 2005
Sequence vs. Series
a1=1
s1=1
a6=6
s6=15+6=21
a2=2
s2=1+2=3
a7=7
s7=21+7=28
a3=3
s3=3+3=6
a8=8
s8=28+8=36
a4=4
s4=6+4=10
a9=9
s9=36+9=45
a5=5
s5=10+5=15
a10=10
s10=45+10=55
Calculus, Section 12.2, Todd Fadoir, CASA, 2005

3. Sequence, Finite Series, Infinite Series
Calculus, Section 12.2, Todd Fadoir, CASA, 2005

4. Convergent Series & Divergent Series
If s exists, we say the series is convergent (or we say it converges)
Otherwise, we say the series is divergent (or we say it diverges)
Calculus, Section 12.2, Todd Fadoir, CASA, 2005

5. Divergent series example
Calculus, Section 12.2, Todd Fadoir, CASA, 2005

6. Simple Tests for Divergence
If limn->∞an≠0, then s diverges.
If limn->∞an=0, then s may converge.
Calculus, Section 12.2, Todd Fadoir, CASA, 2005

7. Convergent series example
Calculus, Section 12.2, Todd Fadoir, CASA, 2005

8. Calculus, Section 12.2, Todd Fadoir, CASA, 2005
Example: Convergent
Calculus, Section 12.2, Todd Fadoir, CASA, 2005

9. Calculus, Section 12.2, Todd Fadoir, CASA, 2005

10. Calculus, Section 12.2, Todd Fadoir, CASA, 2005

11. Calculus, Section 12.2, Todd Fadoir, CASA, 2005
Geometric series
If there is a geometric series with -1<r<1, then the series converges
Otherwise, it diverges
Calculus, Section 12.2, Todd Fadoir, CASA, 2005

12. Calculus, Section 12.2, Todd Fadoir, CASA, 2005

13. Calculus, Section 12.2, Todd Fadoir, CASA, 2005
Geometric series
If there is a geometric series with -1<r<1, then the series converges | to a/(1-r)
Otherwise, it diverges
Calculus, Section 12.2, Todd Fadoir, CASA, 2005

14. Calculus, Section 12.2, Todd Fadoir, CASA, 2005
Example
Calculus, Section 12.2, Todd Fadoir, CASA, 2005

15. Calculus, Section 12.2, Todd Fadoir, CASA, 2005

16. Calculus, Section 12.2, Todd Fadoir, CASA, 2005
Exercises
Section 12.2: 1, 3, 9-31 odd
Calculus, Section 12.2, Todd Fadoir, CASA, 2005