MAT137Y1 – LEC0501
Calculus!
Taylor series
March 25 th , 2019
Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Mar 25, 2019 1
For next lecture
For Wednesday (Mar 27), watch the videos:
• Analytic functions: 14.7, 14.8
• New power series: 14.9, 14.10
Taylor polynomial of a polynomial
Let 𝑓 (𝑥) = 𝑥 3
1 Write the 2nd Taylor polynomial 𝑃 2 for 𝑓 at 0
2 Write the 3rd Taylor polynomial 𝑃 3 for 𝑓 at 0
3 Write the 2nd Taylor polynomial for 𝑓 at 1.
4 Write the 3rd Taylor polynomial for 𝑓 at 1
Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Mar 25, 2019 3
Taylor polynomial of a polynomial
Let 𝑓 (𝑥) = 𝑥 3
1 Write the 2nd Taylor polynomial 𝑃 2 for 𝑓 at 0
2 Write the 3rd Taylor polynomial 𝑃 3 for 𝑓 at 0
3 Write the 2nd Taylor polynomial for 𝑓 at 1.
4 Write the 3rd Taylor polynomial for 𝑓 at 1
Interval of convergence
You have learned the Maclaurin series for the following functions
𝑓 (𝑥) = 𝑒 𝑥 , 𝑔(𝑥) = sin 𝑥, ℎ(𝑥) = cos 𝑥 Compute the interval of convergence of each of these three series.
Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Mar 25, 2019 4
Taylor series not at 0
Write the Taylor series...
1 for 𝑓 (𝑥) = 𝑒 𝑥 at 𝑎 = 2
2 for 𝑔(𝑥) = sin 𝑥 at 𝑎 = 𝜋 4
3 for 𝐻(𝑥) = 1
𝑥 at 𝑎 = 3
You can do these problems in two ways:
• Method 1: Compute the first few derivatives, guess the pattern (and prove it by induction).
• Method 2: Use the substitution 𝑢 = 𝑥 − 𝑎 and
An interesting power series - Homework
Denote by 𝑎 𝑛 the 𝑛-th decimal digit of 𝜋.
Define the power series 𝑆(𝑥) =
∞
∑ 𝑛=1
𝑎 𝑛 𝑥 𝑛 .
Find the radius of convergence and the interval of convergence of 𝑆.
Hint?
𝜋=3. 14 159 26 535 897 932 3846 264 3383 27950
28841971693993751058209749445923078164062862089986280348253421170679So 𝑎 1 = 1, 𝑎 2 = 4, 𝑎 3 = 1, 𝑎 4 = 5 …
Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Mar 25, 2019 6
Hint (to be continued)
3 .1
15 4 653 92
97 58
2 93 8 3 4 6 2 6
43 38 3 27950 288 41 97 1
6 9 3 9 9
51 37
20 058
49 97
59 44
3 2
0 7
8 1 6 40
62 86 2 0899862 80 3
4825342
11
067
214 798
80 65 8 28 13 23
76460
09384460955058223172 5359
40812
848111
5074
102 284
70
38 19
11 52 4622649498955549053038
1964428810975665933 44 61284756
48
233786783165271201
9091
45648566
923 46034861045 4326
64821339360726024914127372 45870066063155881748
81520920962829
254
091715 36436 7892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074
462379962749567351885752724891227938183011949
Hint (to be continued)
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148 0865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756 6593344612847564823378678316527120190914564856692346034861045432664821339360726024914127372458700660631558 8174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092 1861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949 4639522473719070217986094370277053921717629317675238467481846766940513200056812714526356082778577134275778 9609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774 7713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311 8817101000313783875288658753320838142061717766914730359825349042875546873115956286388235378759375195778185 7780532171226806613001927876611195909216420198938095257201065485863278865936153381827968230301952035301852 9689957736225994138912497217752834791315155748572424541506959508295331168617278558890750983817546374649393 1925506040092770167113900984882401285836160356370766010471018194295559619894676783744944825537977472684710 4047534646208046684259069491293313677028989152104752162056966024058038150193511253382430035587640247496473 2639141992726042699227967823547816360093417216412199245863150302861829745557067498385054945885869269956909 2721079750930295532116534498720275596023648066549911988183479775356636980742654252786255181841757467289097 7772793800081647060016145249192173217214772350141441973568548161361157352552133475741849468438523323907394 1433345477624168625189835694855620992192221842725502542568876717904946016534668049886272327917860857843838 2796797668145410095388378636095068006422512520511739298489608412848862694560424196528502221066118630674427 8622039194945047123713786960956364371917287467764657573962413890865832645995813390478027590099465764078951 2694683983525957098258226205224894077267194782684826014769909026401363944374553050682034962524517493996514 3142980919065925093722169646151570985838741059788595977297549893016175392846813826868386894277415599185592 5245953959431049972524680845987273644695848653836736222626099124608051243884390451244136549762780797715691 4359977001296160894416948685558484063534220722258284886481584560285060168427394522674676788952521385225499 5466672782398645659611635488623057745649803559363456817432411251507606947945109659609402522887971089314566 9136867228748940560101503308617928680920874760917824938589009714909675985261365549781893129784821682998948 7226588048575640142704775551323796414515237462343645428584447952658678210511413547357395231134271661021359 6953623144295248493718711014576540359027993440374200731057853906219838744780847848968332144571386875194350 6430218453191048481005370614680674919278191197939952061419663428754440643745123718192179998391015919561814 6751426912397489409071864942319615679452080951465502252316038819301420937621378559566389377870830390697920 7734672218256259966150142150306803844773454920260541466592520149744285073251866600213243408819071048633173 4649651453905796268561005508106658796998163574736384052571459102897064140110971206280439039759515677157700 4203378699360072305587631763594218731251471205329281918261861258673215791984148488291644706095752706957220 9175671167229109816909152801735067127485832228718352093539657251210835791513698820914442100675103346711031 4126711136990865851639831501970165151168517143765761835155650884909989859982387345528331635507647918535893 2261854896321329330898570642046752590709154814165498594616371802709819943099244889575712828905923233260972 9971208443357326548938239119325974636673058360414281388303203824903758985243744170291327656180937734440307 0746921120191302033038019762110110044929321516084244485963766983895228684783123552658213144957685726243344 1893039686426243410773226978028073189154411010446823252716201052652272111660396665573092547110557853763466 8206531098965269186205647693125705863566201855810072936065987648611791045334885034611365768675324944166803 9626579787718556084552965412665408530614344431858676975145661406800700237877659134401712749470420562230538 9945613140711270004078547332699390814546646458807972708266830634328587856983052358089330657574067954571637 7525420211495576158140025012622859413021647155097925923099079654737612551765675135751782966645477917450112 9961489030463994713296210734043751895735961458901938971311179042978285647503203198691514028708085990480109 4121472213179476477726224142548545403321571853061422881375850430633217518297986622371721591607716692547487 3898665494945011465406284336639379003976926567214638530673609657120918076383271664162748888007869256029022 847210403172118608204190004229661711963779213375751149595015660496318629472654736
Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Mar 25, 2019 8
