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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
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