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POWERSERIES
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Maclaurin and Taylor Polynomials
approximation& Sigma notation
The nthTaylor polynomial for fabout x= x0:
n
o
n
on xxn
xf
xx
xf
xx
xf
xxxfxfxP )(!
)(
)(!3
)(
)(!2
)(
))(()()(
0
)(3
0
02
0
0
00
)(!
)()(
!3
)()(
!2
)(
))(()()(!
)(
0
)(3
002
00
000
0
0
)(
n
o
n
o
kn
k
k
xxn
xfxx
xfxx
xf
xxxfxfxxk
xf
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The nthMaclaurin polynomial :
nn
n xn
fx
fx
fxffxP
!
)0(
!3
)0(
!2
)0()0()0()(
)(32
!
)0(
!3
)0(
!2
)0()0()(
!
)0( )(320
0
)(n
nk
n
k
k
xn
fx
fx
fxfxfx
k
f
What is the different between Taylor & Maclaurin series
?
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1kk
u
n
k
knn uuuuus
uuus
uus
us
1
321
3213
212
11
The individual sumsGiven
snis called the nthpartial sumof
the series
1nns is called the sequence of part ialsums.
Meanwhile
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1
321
k
kk uuuuu
Definition
An infinite seriesis an expression thatcan be written in the
form
,...,, 321 uuuThe numbers are called the terms of
the series.
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1. Harmonic series
1 5
1
4
1
3
1
2
11
1
k k
Example diverging series :
2. p-series
ppp
kp kk
1
3
1
2
11
1
1
10 p
3. Geometric Series
0,12
1
aarararaar k
k
k
1r .1 r
a
and its sum is
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Convergence test
i. Alternating series test
4321
1
1)1( aaaaak
k
k
4321
1
)1( aaaaak
k
k
kaaaa 321
0lim
k
k
a
An alternating series converges if the following
two conditions are satisfied:
(b)
(a)
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Absolute Convergence
k
k
k uuuu 211
k
k
k uuuu 211
A series
is said to converge absolutelyif the series of
absolute values
converges
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a series that converge but does not convergeabsolutely is said
converge conditionally
(or to be conditionally convergent)
Note:
k
k 1)1(
4
1
3
1
2
11 1
Example :
using alternating series test ----- converge
but does not converge absolutely
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Maclaurin and Taylor Series
0
2
00
00000
)(
)(!2
)())(()()(
!
)(
k
kk
xxxf
xxxfxfxxk
xf
kk
xxk
xf)(!
)(00
)(
0
)(2
)(
!
)0(
!2
)0()0()0(
!
)0(
k
kk
kk
x
k
fx
fxffx
k
f
Identify which one is the Maclaurin /
Taylor Series?
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!!21
!
2
0 kxxx
kx
k
k
k
The Maclaurin :
ex =
)!12()1(!7!5!3)!12()1(12753
0
12
k
xxxxxk
x kk
k
kk
xsin =
)!2()1(!6!4!21)!2()1(
2642
0
12
k
xxxx
k
x kk
k
k
kxcos =
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Power series in x
kk
k
kk xcxcxccxc 22100
Where ,,, 210 ccc are constants and xis a
variable
Example :
32
0
1 xxxxk
k
!3!2
1
!
32
0
xxx
k
x
k
k
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Radius and interval of convergence
),(
Interval of convergence= the convergence set of a
power series in x.
A series has:Radius of convergence 0= the convergence set is
the
single value x= 0,orRadius of convergence + = the convergence
set is
or
Radius of convergence R= the convergence set extendsbetween Rand
R.
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0
Diverges Diverges
0
Diverges Diverges
Converges
Converges
Radius of convergence R= 0
Radius of convergence R=
Radius of convergence R-R R
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k
kk u
u1lim
so the series converges absolutely if1|| x
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You try this!
Joseph Fourier (1768 1830)
Had crazy idea (1807) :Any periodic functioncan be written as
sum
of sines and cosines ofdifferent frequencies
Revision
http://localhost/var/www/apps/conversion/tmp/scratch_2/Formula%204.pdfhttp://localhost/var/www/apps/conversion/tmp/scratch_2/Formula%204.pdf
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