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# Class Note for MATH 250A at UA

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Date Created: 02/06/15

MATH 250a Fall Semester 2007 Section 2 J M Cushing Thursday November 8 httprnathariz0naeducushing250ahtrnl 95 POLSeries i611 35 61 COClx ac2 pg 602 N the coefficients 95 Ms i011 xa n COClx ac2 x a2 the center the coefficients 95 POLSeries i011 xan COClx ac2 x a2 n0 Examples Zix 1xix2ix3m 39 2 3 710 oo 1n 1 1 Z x2n1xx3x5n n20 2n1 3 5 00 1n 1 n 1 2 1 3 1 4 quot21 3 QC 1 X 1 EX 1 X 1 Zx 1 95 POLSeries i611 35 61 COClx ac2 pg 602 n0 Examples iixn 21xix2 ix3 Note missing terms n0nl 2 3 i 1 x2n1xix3ixsm quot202n1 3 5 00 n l 21 Hyx1lx12lx13lx14 n 2 3 4 n1 95 POLSeries i011 xan COClx ac2 x a2 n0 Examples Zi39x 1xi39x2i39x3 quot0 2 339 Centera1 oo 1n 1 1 Z x2n1xx3x5n n20 2n1 3 5 00 1n 1 n 1 2 1 3 1 4 quot21 3 QC 1 X 1 EX 1 X 1 Zx 1 95 Power Series i011 xan 2C0 ClC ac2 x a2 n0 Basic uestions For which values of x does a power series converge To what does a power series converge by this is meant is there an alternative nonpower series formula Note a power series always converges to CO at its center x a 95 POLSeries i011 xan COClx ac2 x a2 n0 W A power series converges on an interval lx al lt R 95 Power Series i011 xan COClx ac2 x a2 n0 Basic Fact A power series converges on an interval lx al lt R and diverges for Ix al gt R for either some number R 2 0 or for R 00 95 Power Series i011 xan COClx ac2 x a2 n0 Basic Fact A power series converges on an interval lx al lt R and diverges for Ix al gt R for either some number R 2 0 or for R 00 Notes R is called the radius of convergence 95 Power Series icn xan COClx ac2 x a2 n0 Basic Fact A power series converges on an interval lx al lt R and diverges for Ix al gt R for either some number R 2 0 or for R 00 Notes R is called the radius of convergence 0 The series might or might not converge at the endpoints a i R of the convergence interval Examples A geometric series with ratio r x2 converges for diverges for A geometric series with ratio r x2 X lt1 X gt1 A geometric series with ratio r x2 X lt1 converges for converges for x lt 2 3 diverges for x gt 2 diverges for gt1 A geometric series with ratio r x2 X converges for lt1 converges for x lt 2 x diverges for x gt 2 diverges for gt1 Radius of convergence R 2 Radius of convergence R 2 In fact z x n20 2 1 i 2 Radius of convergence R 2 In fact 1 n 1 2 x 2 1 f 2 x 2 We have summed the series Radius of convergence R 2 In fact 1 2 2 x 1 2 We have summed the series It is not often possible to do this We ll some methods next chapter 00 1 n n 00 x 3 3 2 x 4 Geometric with ratio r x 3 w 1 n 1 W W 1 Geometric with ratio r 2 33729 3quot 21 3 7 n0 Geometric with ratio r inx3n 21 3 7 n0 x 3 4 for lt1 or Ix 3lt4 Radius of convergence R 4 Here s a way to calculate the radius of convergence for nongeometric series Here s a way to calculate the radius of convergence for nongeometric series 00 Apply the ratio test to the power series 2 cm x a n20 n1 a n1 a n C n1 x a I l x a Cl l Here s a way to calculate the radius of convergence for nongeometric series 00 Apply the ratio test to the power series 2 cm x a n20 n1 a n1 a n C n1 36 61 C n1 C n x al I l x a Cl l Here s a way to calculate the radius of convergence for nongeometric series 00 Apply the ratio test to the power series 2 cm x a n20 n1 anl Cnl x a Cnl an cm x a 0n C a 1 1 If 11m 2K then 11m le al n 00 C I l gt 00 a n n Here s a way to calculate the radius of convergence for nongeometric series 00 Apply the ratio test to the power series 2 cm x a n20 n1 anl Cnl x a Cnl n x al an cm x a 0n c a 1 1 If 11m 2K then 11m le al n 00 C I l gt 00 a n 71 Apply ratio test K Ix al lt l gt convergence Here s a way to calculate the radius of convergence for nongeometric series 00 Apply the ratio test to the power series 2 cm x a n20 n1 anl Cnl x a Cnl n x al an cm x a 0n c a 1 1 If 11m 2K then 11m le al n 00 C I l gt 00 a n 71 Apply ratio test K Ix al lt l gt convergence K Ix al gt 1 gt divergence Here s a way to calculate the radius of convergence for nongeometric series 00 Apply the ratio test to the power series 2 cm x a n20 n1 anl Cnl x a Cnl n x al an cm x a 0n c a 1 1 If 11m 2K then 11m le al n 00 C I l gt 00 a n 71 Apply ratio test Ix al lt l K gt convergence lx agtlK gt divergence Here s a way to calculate the radius of convergence for nongeometric series 00 Apply the ratio test to the power series 2 cm x a n20 n1 anl Cnl x a Cnl n x al an cm x a 0n c a 1 1 If 11m 2K then 11m le al n 00 C I l gt 00 a n 71 Apply ratio test Ix al lt l K gt convergence lx agtlK gt divergence Radius of convergence is R UK Here s a way to calculate the radius of convergence for nongeornetric series Radius of Convergence Theorem 910 text C n If 11m 1 K then the radius of convergence of E C x a 11 00 C is R 1 K Note If K200 then R20 If K20 then R200 Examples 00 1 n 1 1 x 1x x2 x3 nzon 2 3 Examples 0 1 1 1 x 1x x2 x3 nzon 2 3 1 n1 11m quot1 11m I l OO Cn I l OO n Examples 0 1 n 1 2 1 x 1x x x nzon 2 3 1 39 lim 1 lim quot n gtoo i n gtoo 3 Examples 0 1 n 1 2 1 x 1x x x nzon 2 3 1 39 lim 1 lim quot n gtoo i n gtoo 3 1 11m n gtoon1 Examples 0 1 n 1 2 1 x 1x x x nzon 2 3 1 39 lim 1 lim quot n gtoo i n gtoo 3 lim L0 n gtoon1 K Examples ml 1 1 x 1x x2 x3 nzon 2 3 1 n1 n 1 11m quot1211m 11m 11m OK n gtoo C n gtoo 1 n gtoon1 n gt radius of convergence is R 2 00 This power series converges for all x Examples 00 n l Hg x uquotx 1gt x 1gt2 x 1gt3 1gt4 1 lim quot1 lim quot1 11mi1K n gtoo C n gtoo l n gtoon1 n gt the radius of convergence is R 1 K 1 Examples 00 n l Hg x uquotx 1gt x 1gt2 x 1gt3 x 1gt4 1 1m1 11m1quot11m1 3 LK n gtoo C n gtoo l n gtoon1 n gt the radius of convergence is R 1 K 1 This power series converges on the interval pqlt1 m 0ltxlt2 Examples i 1 x2nl x 1 3 1 n02nl 3 5 missing terms have coefficients O This implies we cannot calculate 0n1 l C when Cn O which is every other ratio n1 71 Examples 1 x2nl x 1 3 1 0 2n 1 3 5 Apply ratio test directly Examples 1 x2nl x 1 3 1 n02nl 3 5 Apply ratio test directly 1 x2nl 211 1 Examples 1 x2nl x 1 3 1 n02nl 3 5 Apply ratio test directly DHH 2nll x 2 72 1 1 1 x2nl 211 1 a n1 a n Examples 1 x2nl x 1 3 1 n02nl 3 5 Apply ratio test directly DHH 2nll l 2n3 x Ix an1 2nl1l 2n3l an 1n 2 l x 2nl 2nllx 2 1 I Apply ratio test directly i x2n11 1 Ix 2n3 an1 2n11 2n3 2n1x2 1quot M 2m 2m 2n1x a n Examples 1 x2nl x 1 3 1 n02nl 3 5 Apply ratio test directly 1quot1 x2nll a 2l llll 2n3l n1 an 1quot 2m 1 MW 2n3 2nllx 2nl 1232n1 2n3 l232nl2n2 v A l 3 b v Examples 1 x2nl x 1 3 1 n02nl 3 5 Apply ratio test directly 1quot1 x2nll a 2l llll 2n3l n1 an 1 2m 1 MM 2n3 2nllx 2N1 T 2nl 1232n1 1 2n3 l232nl2n22n3 2n22n3 Apply ratio test directly 1 n1 l2 2n22n3x Apply ratio test directly 1 139 0 1 1131100 2n22n3x lt for all x a n1 71 00 a n Examples 00 1 x2n12x 1 3 1 n02nl 3x 5x Apply ratio test directly a l 2 l39 quot1 l39 0 1 1131100 an niIPw2n22n3lxl lt for czle Ratio test implies series converges for all x that is to say R 2 oo Ratio Test doesn t determine what happens at the endpoints of the interval of convergence Ratio Test doesn t determine what happens at the endpoints of the interval of convergence Anything goes at the end points Example oo 1 11 1 2 x 1 has interval of convergence 0 lt x lt 2 n1 n Ratio Test doesn t determine what happens at the endpoints of the interval of convergence Anything goes at the end points Example oo 1 11 1 2 x 1 has interval of convergence 0 lt x lt 2 n1 n At endpoint x O i 1n 1 x 1 i 1n 1 1 n n Ratio Test doesn t determine what happens at the endpoints of the interval of convergence Anything goes at the end points Example oo 1 11 1 2 x 1 has interval of convergence 0 lt x lt 2 n1 n oo 1n 1 n 00 1 211 1 At endp01ntx0i 2 x l 2 n1 nl n n Ratio Test doesn t determine what happens at the endpoints of the interval of convergence Anything goes at the end points Example oo 1 11 1 2 x 1 has interval of convergence 0 lt x lt 2 n1 n 2n 1 00 1 11 1 00 1 Atendpointx0i x1n2 n21 quot1 n odd integer power Ratio Test doesn t determine what happens at the endpoints of the interval of convergence Anything goes at the end points Example 00 ln 1 n 2 x 1 has interval of convergence 0 lt x lt 2 n1 n 00 1 11 1 00 1 Atendpointx0i 35 1 71 1 n n21 n Ratio Test doesn t determine what happens at the endpoints of the interval of convergence Anything goes at the end points Example 00 ln 1 n 2 x 1 has interval of convergence 0 lt x lt 2 n1 n 00 1 11 1 00 1 00 1 Atendpointx0i x l nl n n21 n n21 n divergent pseries Ratio Test doesn t determine what happens at the endpoints of the interval of convergence Anything goes at the end points Example oo 1 11 1 2 x 1 has interval of convergence 0 lt x lt 2 n1 n At endpoint x O series diverges Ratio Test doesn t determine what happens at the endpoints of the interval of convergence Anything goes at the end points Example oo 1 11 1 2 x 1 has interval of convergence 0 lt x lt 2 n1 n At endpoint x O series diverges 00 1 n1 00 1 11 1 Atendpointx2i x1nZ nl quot 1 n n Ratio Test doesn t determine what happens at the endpoints of the interval of convergence Anything goes at the end points Example oo 1 11 1 2 x 1 has interval of convergence 0 lt x lt 2 nl n At endpoint x O series diverges 00 1 n1 00 1 11 1 Atendpointx2i x1nZ nl quot1 n n convergent alternating series Ratio Test doesn t determine what happens at the endpoints of the interval of convergence Anything goes at the end points Example oo 1 11 1 2 x 1 has interval of convergence 0 lt x lt 2 n1 n At endpoint x O series diverges At endpoint x 2 series converges Ratio Test doesn t determine what happens at the endpoints of the interval of convergence Anything goes at the end points Example 1n 1 1 n M8 x 1 has interval of convergence 0 lt x lt 2 n oo 1 11 1 2 x 1 has domain of convergence 0 lt x S 2 n1 n Chapter 101 Taylor Polynomials Polynomials are easy With which to work differentiate integrate etc For this reason is often convenient to approximate other functions by polynomials First Degree Polynomial Approximation Approximate y f x at x a by a first degree polynomial Geometrically approximate graph of f x by that of its tangent at x a First Degree Polynomial Approximation Approximate y f x at x a by a first degree polynomial Geometrically approximate graph of f x by that of its tangent at x a y a f 61 First Degree Polynomial Approximation Approximate y f x at x a by a first degree polynomial Geometrically approximate graph of f x by that of its tangent at x a y pX fa f ax a a f 61 First Degree Polynomial Approximation Approximate y f x at x a by a first degree polynomial Geometrically approximate graph of f x by that of its tangent at x a y pX fa f39ax a a f 61 Passes through same point as f x x I First Degree Polynomial Approximation Approximate y f x at x a by a first degree polynomial Geometrically approximate graph of fx by that of its tangent at x a y pX fa f ax a a f 61 Passes throbvme point as f x a x Has same slope as f x First Degree Polynomial Approximation 1910 faf39aX a for x z a First degree Taylor polynomial of f x at x a First Degree Polynomial Approximation 1910 faf39aX a for x z 61 Examples fxex at a0 First Degree Polynomial Approximation 1910 faf39aX a for x z 61 Examples fxex at a0 f0 1 f39x ex First Degree Polynomial Approximation 1910 faf39aX a for x z 61 Examples fxex at a0 f0 1 f 39x ex 2 f 0 1 First Degree Polynomial Approximation p1ltxgt fa f39ltagtx a z fx for x z a Examples fxex at a0 f0 1 f 39x ex gt f 0 1 p1 x 2 1 x First Degree Polynomial Approximation 1910 faf39aX a for x z 61 Examples fxex at a0 f0 1 f 39x ex 2 f 0 1 p1xlxex for sz

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