Taylor and Maclaurin Series

on an open intervalAssume FC x an Cx al containing au O

Q Can we express Ca in terms of F 2

O at x aObservations 0 at x a

r lL l

x af 3 x a2 dd z

K a 3 3.2 Cx a

3 4d d

x af 3.2 I 31 k al Od x

Moregenerally o iz k t nk

k al Ia

n it k u

I k

7 a T dCk k

gL Cuca a In o K a

term byterm teth derivative

Conclusiontext TLan Ca al on open intervalu o

autaing aan 7 Cal tu au n I

I o l

Definition

Taylor series AI ca

FCK centered Halt al ex aft

at x a T 7 caL t

CK al calledn o Maclaurin

SeriesN L

Special Cad a o T f coL siu n l

Importanttext Icu Cx al an open interval containing a

u o

Tcu a Taylor series at 7 x at x a

Examples C l l

Maclaurin

yLIt x x x T in Series A

LC l l

4 0 I

s

z lull 1 1 Ex Tai FEELLLu I lull 1 1C l lN 2 ut

I arctanbel a Te n Maclaurin1 zu series rtn o arctan a

We can of course determine Taylor series by direct

calculation

Examplest7cal e 7 a e 7 ol I

Fw all a o

Maclaurin Series T an I x 1 1of ex 1 n I

n o

1

y text sin x

7 o o F co I This pattern then7 o o 7 co repeatsMaclaurin seriesof sink II 1

y 1 Gc say

1 co I 7 Col 0 This pattern tenc7 Col I 1 Col o repeats

Maclaurin seriesof c six

I 1

Ly Binomial series

text i ykT any real number

k l7 Cal le Cita I'Col k7 x k Ck i xp 7 o teck i

R n 117 x k k i K nti Ita

7 o k Ck i CK u i

Notation ku kCk i Ck

u

Maclaurin series T Cn xA Chalk L

I

W u There is no guarantee ai general

that a function equals its Taylor series on an

open interval containg a Whether it does actuallydepends quite deeply on the complex numbers

Important Question When is 7Cx equal to

its Tapu series centered at a

Nth Taylor polynomialTN 4 7 a t ex alt ca a

TT7 ca InkaL ex ai E II ca alU I 4 0

Nth remainder Lim TN saN so

Let Riv Cal 1 Gel Tuckso

T 7 catext L x al text Liz IN GYU I

ILim R Lx O t being approximatedN a big polynomials to

higher and higheraccuracy

ie tu Lage N

Treas Tu teas

Taylors Inequality M o and d oLethtt

IF I t ca f E M tu au x in

a d a d C Cen An upper bound on

Nth remainderC N t I

I Rascal E Ix alN 1 I

f ou all x in a d at d

Example f Cx sin x a o d o

y Is in cx1 s x

htt

II ca I E I In all in C did

Io El RN x e 1 1 Fru alla in C did

Ratio TestT1 1,1

lol convergent fuiyj.ua lxl toN

Squeeze theorem

Lim Rn Cx I oN s

siuc.cl x 1 t 7ham x

in C didBut d o was arbitrary Amazing Fact

Isiuc.cl x 1 two

H

sincx can be represented to higher and higher accuracy

using Taylor polynomials Tzu 1164

Awesome

Importantspeciaseson C a a

e l t a t t t

on C o as

sina.ci Ex Ifton C a 1c sext I 3 If t

Ion C l i 1l t se t si t is t

on C l l

Iu Ctx t

ou L l lare tan x x c

C t l

l x It textk k

I kCk k x's

Example J e da

Problem Cannot find antiderivatve by anyp method

u L a aa

e I x 27 337 T IL n In o

you C a 1

L C v2 3e

2I c oil tE y

a

I of z Tei3 L uln o

Je de C t az II t

IConstant T n Intlset C DIntegration Cuti ut

U