Chabot Mathematics
§10.3 Series:
Power & Taylor
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
Review § 10.2
 Any QUESTIONS About
• §10.2 Convergence Tests
 Any QUESTIONS About
HomeWork
• §10.2 → HW-18
Chabot College Mathematics
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
§10.3 Learning Goals
 Find the radius and interval of
convergence for a power series
 Study term-by-term
differentiation and
integration of
power series
 Explore Taylor
series representation
of functions
Chabot College Mathematics
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
Power Series
 General Power Series:
n 
a0 x  a1 x  a2 x  a3 x   ak x     an x n
0
1
2
3
k
n 0
• A form of a GENERALIZED POLYNOMIAL
 Power Series Convergence Behavior
• Exclusively ONE of the following holds True
a) Converges ONLY for x = 0 (Trival Case)
b) Converges for ALL x
c) Has a Finite “Radius of Convergence”, R
Chabot College Mathematics
4
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
Radius of Convergence
 For the General Power Series
n 
a0 x  a1 x  a2 x  a3 x   ak x     an x n
0
1
2
3
k
n 0
 Unless a power series converges at any
real number, a number R > 0 exists
such that the series CONverges
absolutely for each x such that | x | < R
and DIverges for any other x
 Thus the “Interval of Convergence”
 R  xconv
Chabot College Mathematics
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 R
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
Example  Radius of Conv.
 Find R for the Series:
• Radius of Convergence

k k
x

k
k 0 4
• Interval of Convergence
 SOLUTION
 Use the Ratio Test
(k +1)x 4
lim
k®¥
kxk 4 k
k+1
Chabot College Mathematics
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k+1
x(k +1)
<1
<1 ¥ lim
k®¥
4k
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
Example  Radius of Conv.
 Continue with Limit Evaluation:
x
x(k +1)

k  1
¥ lim
<1  lim
1
k®¥
4k
4 k  k
x
 1  1
 Thus R = 4
4
 4  x  4
 The Interval of
Convergence

k
k
 Thus This Series
 3

k
Converges
k 0 4
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
Functions as Power Series
 Many Functions can be represented as
Infinitely Long PolyNomials
 Consider this Function and Domain
1
f x  
for x  1
1 x
 Recall one of The Geometric Series
n 
1
 1x 0  1x1  1x 2  1x 3  1x k    1  x n  f  x 
1 x
 Thus
Chabot College Mathematics
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n 0
n 
1
f x  
  xn
1  x n 0
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
Example  Fcn by Pwr Series
2
2x
 Write as a Power Series → f x  
2
4

x
• Also Find the Radius of Convergence
 SOLUTION:
 Start with the
1
=1+ x + x2 + x3 +...
GeoMetric Series 1- x
 First Cast the Fcn into the Form
      
 

2
3
1
p
q
q
q
p
q
Ax p

Ax
1

Bx

Bx

Bx

...

Ax
Bx

1  Bx q
n 0
 
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
n
Example  Fcn by Pwr Series
 Using Algebraic Processes on the Fcn
2
2


2x2
1
2
x
1
x
1
2

f x  
 2x
 

2
2
2
4 x
4 1 x 4  4  1 x 4
2 1 x2 4





 Thus by the Geometric Series
2
2x
x
1
x
f x  


2
2
4 x
2 1 x 4
2
 
 x 4
n
2 n 
2

2
n 0
 Then the Function by Power Series
2
2x
x
 f x  
2
4 x
2
Chabot College Mathematics
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 x 4
n
2 n 
2
n 0
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx

Example  Fcn by Pwr Series
 Now find the Radius of Convergence by
the Ratio Test
 x   1  1 x  lim  x 
2
 x
 x

x   x 
1
 lim
1 
x  lim 
2
 x
1 2
lim x 
k  2
1 2

x
2
1
1
2 k
2 k
4
1 2
x 
2
Chabot College Mathematics
k 
4
k 
1
2
4
1

11
2 k 1
1

1
4
2 k
2 k 1
4
1
2 k
1
4
2 1
2
1
k 
4
1

1 4
x 1  1
4 x  lim 1  1 
k 
8
2

2
x
1
4
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
Example  Fcn by Pwr Series
 Thus for Convergence
1 4

8 x 1  1  x 4  8  x   4 8
8

 So the Interval of Convergence:
4 8 
x  4 8
 And also the Radius of Convergence
R4 8
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
Pwr Series Derivatives & Integrals
 Consider a Convergent Power Series
n 
a0  a1 x  a2 x  a3 x     an x n
2
3
x R
converging for
n 0
 And an Associated Function
n 
f x    an x n with Domain  R  x  R
n 0
 If f(x) is differentiable over −R<x<R, then
n 
d 
2
3
n
 f x   a0  a1 x  a2 x  a3 x     an x 
dx 
n 0

n 
df x 
2
3
 a1  2a2 x  3a3 x  4a4 x    nan x n 1
dx
n 1
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
Pwr Series Derivatives & Integrals
 If f(x) is Integrable over −R<x<R, then
n 

2
3
n
an x   dx
  f x   a0  a1 x  a2 x  a3 x    
n 0

n 

2
3
n
a0  a1 x  a2 x  a3 x    dx     an x   dx
 n 0

 f x  dx   


n 
 x n1 
x2
x3
x4
  C
f x   dx  C  a0 x  a1  a2  a3     an 
2
3
4
n 0
 n 1
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
Pwr Series Derivatives & Integrals
 Thus the Derivative of a df n
n 1
  nan x
Power-Series Function
dx n 1
 Thus the AntiDerivative of a
Power-Series Function

Chabot College Mathematics
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 x n1 
  C
f x   dx   an 
n 0
 n 1
n 
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
Example  Find Fcn by Integ
 Find a Power Series Equivalent for
f x  2x  2 ln 2  x  2 ln x  2
 SOLUTION:
d
d
 First take: f x    2 x  2 ln 2  x   2 ln 2  x 
dx
dx
2
2
2
2x
= -2 (-1) +
=
2- x
x + 2 4 - x2
 Recognize
from Before
Chabot College Mathematics
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2
2
2
2x
x
1
x


2
2
4 x
2 1 x 4
2
 

x 

 
n 0  4 

2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
n
Example  Find Fcn by Integ
 Recover the Original Fcn by taking the
AntiDerivative of the Just Determined
Derivative
2
2 
2 n

x  
2x
x
 df x 
dx 
dx 
dx  f x 
 

dx 
 x2
f x    
2
 4 x
2
  2   4 

n 0

 x2 
x2
0  4    dx   2  1 
 


1
 
x 
2
4



1
4
x

2 2
 x 2 x 4 x 6 x8

f x       
   dx
 2 8 32 128

Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx

 ...  dx
Example  Find Fcn by Integ
1 3 1 5
1 7
 Then f x   C  x  x 
x  ...
6
40
224
 To Find C use the original Function
f 0  2  0  2 ln 2  0  2 ln 0  2  0  2ln 2  ln 2  0
 Use f(0) = 0 in Power Series fcn
1 3 1 5
1 7
f 0  C  0  0 
0  ...  0  C  0
6
40
224
 Then the Final Power Series Fcn

1 3 1 5
1 7
x 2 k 3
f x   x  x 
x  ...  
k


6
40
224
2
2
k

3

4
k 0
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
Taylor Series
 Consider some general Function, f(x),
that might be Represented by a Power
n 
2
3
Series f x   a0  a1 x  a2 x  a3 x     an x n
n 0
 Thus need to find CoEfficients, an, such
that the Power Series Converges to f(x)
over some interval. Stated
Mathematically Need an so that:
n 
converges
n
a
x

 n  f x  for
x R
n 0
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
Taylor Series
 If x = 0 and if f(0) is KNOWN then
f 0  a0  a1 0  a2 0 2  a3 03    a0  0  a0  f 0
• a0 done, 1→∞ to go….
 Next Differentiate Term-by-Term
n 
df x 
2
3
dx
 a1  2a2 x  3a3 x  4a4 x    nan x n 1
n 1
 Now if the First Derivative (the Slope) is
KNOWN when x = 0, then
df
dx
df
 a1  2a2 0  3a3 0  4a4 0   a1  0  a1 
dx
2
x 0
Chabot College Mathematics
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3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
x 0
Taylor Series
 Again Differentiate Term-by-Term
n 
d2 f
2
n2



2
a

3

2
a
x

4

3
a
x


n
n

1
a
x

2
3
4
n
2
dx
n2
 Now if the 2nd Derivative (the Curvature)
is KNOWN when x = 0, then
d2 f
dx 2
x 0
2
d
f
2
 2a2  3  2a3 0  4  3a4 0   2a2  0  2a2  2
dx
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
x 0
Taylor Series

Another
Differentiation
3
n 
d f
2
n 3




3

2
a

4

3

2
a
x

5

4

3
a
x


n
n

1
n

2
a
x

3
4
4
n
3
dx
n 3
d3 f
dx 3
 Again if the 3rd Derivative is KNOWN at
x=0
x 0
3
d
f
2
 3  2a3  4  3  2a4 0  5  4  3a4 0   6a3  0  6a3  3
dx
 Recognizing
the Pattern:
Chabot College Mathematics
22
d n  f
n!an  n 
dx
 an 
x 0
d n  f
dx n 
n!
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
x 0
x 0
Taylor Series
 Thus to Construct a Taylor (Power)
Series about an interval “Centered” at
x = 0 for the Function f(x)
• Find the Values of ALL the Derivatives of
f(x) when f(x) = 0
n 
• Calculate the Values of the
Taylor Series CoEfficients by an 
• Finally Construct the
Power Series from
the CoEfficients
Chabot College Mathematics
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d f
dx n 
n!
n 
f  x    an x n
n 0
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
x 0
Example  Taylor Series for ln(e+x)
 Calculate the Derivatives
df dx
d
1
lne  x  
dx
e x
d 2 f dx 2
d  1 
1



dx  e  x  e  x 2
d 3 f dx 3
d  1 
2


2
dx  e  x   e  x 3
 Find the Values of the Derivatives at 0
df
dxx 0
1
1

e0 e
Chabot College Mathematics
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d
2

f dx 2 x 0
1
1
 2
2
e  0 e
d
3

f dx 3 x 0
2
2
 3
3
e  0 e
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
Example  Taylor Series for ln(e+x)
 Generally d n  f dx n  x 0   1 nn  1! for n  1
e
 Then the CoEfficients
n 1
an

d

n 
f dx
n!
n 

x 0
 1n1 n  1!

n
e
n!

 1

n 1
ne
n
for n  1
 The 1st four CoEfficients
ln(e+ 0)
1 (e+ 0) 1
a0 =
=1
a1 =
=
0!
1!
e
-1 (e+ 0)2
1
2 (e+ 0)3
1
a2 =
= - 2 a3 =
= 3
2!
2e
3!
3e
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
Example  Taylor Series for ln(e+x)
 Then the Taylor Series

ln e  x   a0 x 0   an x n
n 1
n 1

 n



1
0
 a0 x   
x
n 
n 1  n  e


(1) n 1 x n
ln(e  x)  1  
n
ne
n 1

Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
Taylor Series at x ≠ 0
 The Taylor Series “Expansion” can
Occur at “Center” Values other than 0
 Consider a function
n 
n




f
x

a
x

b
stated in a series

n
n 0
centered at b, that is:
 Now the the Radius of Convergence for
the function is the SAME as before:
x  b   R

 R  x b  R

 R b  x bb  R b   R b  x  R b
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
Taylor Series at x ≠ 0
 To find the CoEfficients
n 
d
f
need (x−b) = 0 which
dx n  x b f n  b 
requires x = b, Then the an 

n!
n!
CoEfficient Expression
 The expansion about non-zero centers
is useful for functions (or the
derivatives) that are NOT DEFINED
when x=0
• For Example ln(x) can NOT be expanded
about zero, but it can be about, say, 2
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
Example  Expand x½ about 4
 Expand about b = 4: f x  
x
 The 1st four Taylor CoEfficients
4
a0 =
=2
0!
1 -1/2
(4)
1
2
a1 =
=
1!
4
1 -3/2
- (4)
1
4
a2 =
=2!
64
3 -5/2
(4)
1
8
a3 =
=3!
512
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
Example  Expand x½ about 4
 SOLUTION:
 Use the CoEfficients to Construct the
Taylor Series centered at b = 4

x   an ( x  b) n
n 0
1
1
1
5
2
3
= 2 + (x - 4) - (x - 4) +
(x - 4) (x - 4)4 +...
4
64
512
16384
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30
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
Example  Expand x½ about 4
 Use the Taylor Series centered at b = 4
to Find the Square Root of 3

4
n 0
n 0
3   an (3  b) n   an (3  b) n
1
1
1
5
2
3
 2  (3  4)  (3  4) 
(3  4) 
(3  4) 4
4
64
512
16384
1
1
1
5
2
3
 2  (1)  (1) 
(1) 
(1) 4
4
64
512
16384
 2  0.25  0.0156  0.0020  0.0003
 2  0.2679  1.7321 By MATLAB  1.7320508
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
WhiteBoard PPT Work
 Problems From §10.3
• P39 → expand about f  x   ln x
b = 1 the Function
x
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
All Done for Today
Brook
Taylor
(1685-1731)
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
2a
–
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34
BMayer@ChabotCollege.edu
2b
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
P10.3-39 Taylor Series
 Da1 := diff(ln(x)/x, x)
 Db2 := diff(Da1, x)
 Dc3 := diff(Db2, x)
 Dd4 := diff(Dc3, x)
Chabot College Mathematics
36
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
P10.3-39 Taylor Series
 ln(x)/x, x
 f0 := taylor(ln(x)/x, x = 1, 0)
 f1 := taylor(ln(x)/x, x = 1, 1)
 f2 := taylor(ln(x)/x, x = 1, 2)
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
P10.3-39 Taylor Series
 f3 := taylor(ln(x)/x, x = 1, 3)
 f4 := taylor(ln(x)/x, x = 1, 4)
 d6 := diff(ln(x)/x, x $ 5)
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
P10.3-39 Taylor Series

plot(f0, f1, f2, f3, f4, f5, x =0.5..3, GridVisible = TRUE,
LineWidth = 0.04*unit::inch,
Width = 320*unit::mm, Height = 180*unit::mm,
AxesTitleFont = ["sans-serif", 24],
TicksLabelFont=["sans-serif", 16])
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BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
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BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
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BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx
Chabot College Mathematics
42
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BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx