MTH 253 Calculus (Other Topics)Sums of the Terms of a SequenceInfinite SeriesPartial Sums of an Infinite SeriesConverging/Diverging SeriesGeometric SeriesSlide 7Slide 8Slide 9Slide 10The Harmonic SeriesSlide 12MTH 253Calculus (Other Topics)Chapter 10 – Infinite SeriesSection 10.3 – Infinite SeriesCopyright © 2006 by Ron Wallace, all rights reserved.Sums of the Terms of a Sequence 1310nn�=� �=� ��0.3, 0.03, 0.003, 0.0003, ...Adding these terms gives …10.3333...3=1910nn�=� �� ��What would be the sum of the terms of ?Infinite SeriesThe sum of the terms of a sequence is called an infinite series.Notation:1 2 3 41...kku u u u u�== + + + +�NOTES:• uk is some function of k whose domain is a set of integers.• k can start anywhere (0 or 1 is the most common)Partial Sums of an Infinite Series1 2 3 41...kku u u u u�== + + + +�2 1 2s u u= +3 1 2 3s u u u= + +1 2 31...nn n kks u u u u u== + + + + =�1 1s u={ }1nns�=Sequence of partial sums.●●●Converging/Diverging Series1 2 3 41...kku u u u u�== + + + +�{ }1nns�=Ifconverges to ,Sthen the series converges and1kku S�==�If the sequence of partial sums diverges, then so does the series (it has no sum).S is not often easy or even possible to determine!Geometric SeriesEach term is obtained by multiplying the proceeding term by a fixed constant.2 30...kkar a ar ar ar�== + + + +�Example:1 2103 3 3 1 3 1...10 10 10 10 10 10kk�+=� � � �= + + +� � � �� � � ��NOTE: w/ geometric series, k usually starts with 0 (not required)310a =110r =Geometric SeriesUnder what conditions does a geometric series converge?0kkar�=�Case 1a: r = 10 0... kk knar a a a as� �= == = + + +=� �( 1)a n +limnns��=�Divergent!Geometric SeriesUnder what conditions does a geometric series converge?0kkar�=�Case 1b: r = -10 0... kk knar a a a as� �= == = - + -=� �, if is even0, if is odda nn���limnns��=DNEDivergent!Geometric SeriesUnder what conditions does a geometric series converge?0kkar�=�Case 2: |r| 12 3...nns a ar ar ar ar= + + + + +2 3 4 1...nnrs ar ar ar ar ar+= + + + + +1nn ns rs a ar+- = -timesrsubtract11(1 )1 1nnna ar as rr r++-= = -- -Geometric SeriesUnder what conditions does a geometric series converge?0kkar�=�Case 2: |r| 1Convergent if |r| < 1; Divergent Otherwise11(1 )1 1nnna ar as rr r++-= = -- -limnns��=, if 11, if 1arrr�<�-��� >�The Harmonic Series11 1 1 11 ...2 3 4kk�== + + + +�12 13 2111213 1n nss ss ss sn-== += +���= +{ }1nns�=is a strictly increasing sequenceConverges if there is an upper bound.The Harmonic Series11 1 1 11 ...2 3 4kk�== + + + +�224 2 2221 1 1 212 2 2 21 1 1 1 33 4 4 4 2 12nss s s sns= + > + == = + + > + + =���+>Diverges because there is no possible upper bound.12nM+>Since for any possible upper bound M, you can find an n where
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