AP Calculus March 22, 2010 Berkley High School, D1B1

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

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

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

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

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

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

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

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

Calculus, Section 12.2, Todd Fadoir, CASA, 2005

Calculus, Section 12.2, Todd Fadoir, CASA, 2005

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

Calculus, Section 12.2, Todd Fadoir, CASA, 2005

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

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

Calculus, Section 12.2, Todd Fadoir, CASA, 2005

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