ENGG2013 Unit 21
Power Series
Apr, 2011.
Charles Kao
• Vice-chancellor of CUHK
from 1987 to 1996.
• Nobel prize laureate in
2009.
K. C. Kao and G. A. Hockham, "Dielectric-fibre surface waveguides
for optical frequencies," Proc. IEE, vol. 133, no. 7, pp.1151–1158, 1966.
“It is foreseeable that glasses with a bulk loss of
about 20 dB/km at around 0.6 micrometer will be obtained,
as the iron impurity concentration may be reduced to 1 part per million.”
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Special functions
From the first paragraph of Prof. Kao’s paper
(after abstract), we see
• Jn = nth-order Bessel function of the first kind
• Kn = nth-order modified Bessel function of the
second kind.
• H(i)= th-order Hankel function of the ith
type.
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J(x)
• There is a parameter  called the “order”.
• The th-order Bessel function of the first kind
– http://en.wikipedia.org/wiki/Bessel_function
• Two different definitions:
– Defined as the solution to the differential
equation
– Defined by power series:
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Gamma function (x)
• Gamma function is the extension of the
factorial function to real integer input.
– http://en.wikipedia.org/wiki/Gamma_function
• Definition by integral
• Property : (1) = 1, and
for integer n, (n)=(n – 1)!
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Examples
• The 0-th order Bessel function of the first kind
• The first order Bessel function of the first kind
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INFINITE SERIES
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Infinite series
• Geometric series
– If a = 1 and r= 1/2,
=1
– If a = 1 and r = 1
1+1+1+1+1+… diverges
– If a = 1 and r = – 1
1–1+1–1+1–1+…
– If a = 1 and r = 2
1+2+4+8+16+… diverges
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diverges
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Formal definition for convergence
• Consider an infinite series
– The numbers ai may be real or complex.
• Let Sn be the nth partial sum
• The infinite series is said to be convergent if there is a
number L such that, for every arbitrarily small  > 0, there
exists an integer N such that
• The number L is called the limit of the infinite series.
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Geometric pictures
Complex infinite series
Real infinite series
Im
Complex plane
S2 S1
S0
L

L-
Re
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L
L+
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Convergence of geometric series
• If |r|<1, then
is equal to
converges, and the limit
.
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Easy fact
• If the magnitudes of the terms in an infinite
series does not approach zero, then the
infinite series diverges.
• But the converse is not true.
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Harmonic series
is divergent
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But
is convergent
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Terminologies
• An infinite series z1+z2+z3+… is called
absolutely convergent if |z1|+|z2|+|z3|+… is
convergent.
• An infinite series z1+z2+z3+… is called
conditionally convergent if z1+z2+z3+… is
convergent, but |z1|+|z2|+|z3|+… is
divergent.
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Examples
•
is conditionally convergent.
•
is absolutely convergent.
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Convergence tests
Some sufficient conditions for convergence.
Let z1 + z2 + z3 + z4 + … be a given infinite series.
(z1, z2, z3, … are real or complex numbers)
1. If it is absolutely convergent, then it converges.
2. (Comparison test) If we can find a convergent
series b1 + b2 + b3 + … with non-negative real
terms such that
|zi|  bi for all i,
then z1 + z2 + z3 + z4 + … converges.
http://en.wikipedia.org/wiki/Comparison_test
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Convergence tests
3. (Ratio test) If there is a real number q < 1,
such that
for all i > N (N is some integer),
then z1 + z2 + z3 + z4 + … converges.
If for all i > N ,
, then it diverges
http://en.wikipedia.org/wiki/Ratio_test
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Convergence tests
4. (Root test) If there is a real number q < 1,
such that
for all i > N (N is some integer),
then z1 + z2 + z3 + z4 + … converges.
If for all i > N ,
, then it diverges.
http://en.wikipedia.org/wiki/Root_test
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Derivation of the root test from
comparison test
for all i  N. Then
• Suppose that
for all i  N. But
is a convergent series (because q<1). Therefore
z1 + z2 + z3 + z4 + … converges by the
comparison test.
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Application
• Given a complex number x, apply the ratio test to
• The ratio of the (i+1)-st term and the i-th term is
Let q be a real number strictly less than 1, say q=0.99.
Then,
Therefore exp(x) is convergent for all complex number x.
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Application
• Given a complex number x, apply the root test to
• The ratio of the (i+1)-st term and the i-th term is
Let q be a real number strictly less than 1, say q=0.99.
Then,
Therefore exp(x) is convergent for all complex number x.
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Variations: The limit ratio test
• If an infinite series z1 + z2 + z3 + … , with all
terms nonzero, is such that
Then
1. The series converges if < 1.
2. The series diverges if  > 1.
3. No conclusion if  = 1.
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Variations: The limit root test
• If an infinite series z1 + z2 + z3 + … , with all
terms nonzero, is such that
Then
1. The series converges if < 1.
2. The series diverges if  > 1.
3. No conclusion if  = 1.
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Application
• Let x be a given complex number. Apply the
limit root test to
• The nth term is
• The nth root of the magnitude of the nth term
is
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Useful facts
• Stirling approximation: for all positive integer
n, we have
•
J0(x) converges for every x
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POWER SERIES
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General form
• The input, x, may be real or complex number.
• The coefficient of the nth term, an, may be
real or complex number.
http://en.wikipedia.org/wiki/Power_series
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Approximation by tangent line
2
1.5
1
0.5
y
0
-0.5
-1
-1.5
y = log(x)
Tangent line at x=0.6
-2
-2.5
0
0.2
0.4
0.6
0.8
1
x
1.2
1.4
1.6
1.8
2
x = linspace(0.1,2,50);
plot(x,log(x),'r',x, log(0.6)+(x-0.6)/0.6,'b')
grid on; xlabel('x'); ylabel('y');
legend(‘y = log(x)’, ‘Tangent line at x=0.6‘)
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Approximation by quadratic
1
y = log(x)
Second-order approx at x=0.6
0.5
y
0
-0.5
-1
-1.5
-2
0.2
0.4
0.6
0.8
x = linspace(0.1,2,50);
plot(x,log(x),'r',x, log(0.6)+(x-0.6)/0.6-(x-0.6).^2/0.6^2/2,'b')
grid on; xlabel('x'); ylabel('y')
legend(‘y = log(x)’, ‘Second-order approx at x=0.6‘)
1
1.2
1.4
1.6
1.8
2
x
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Third-order
4
y = log(x)
Third-order approx at x=0.6
3
2
y
1
0
-1
-2
-3
0
0.2
0.4
0.6
0.8
1
x
1.2
1.4
1.6
1.8
2
x = linspace(0.05,2,50);
plot(x,log(x),'r',x, log(0.6)+(x-0.6)/0.6-(x-0.6).^2/0.6^2/2+(x-0.6).^3/0.6^3/3,'b')
grid on; xlabel('x'); ylabel('y')
legend('y = log(x)', ‘Third-order approx at x=0.6')
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Fourth-order
1
0
y
-1
-2
-3
-4
-5
y = log(x)
Fourth-order approx at x=0.6
0
0.2
0.4
0.6
0.8
1
x
1.2
1.4
1.6
1.8
2
x = linspace(0.05,2,50);
plot(x,log(x),'r',x, log(0.6)+(x-0.6)/0.6-(x-0.6).^2/0.6^2/2+(x-0.6).^3/0.6^3/3-(x-0.6).^4/0.6^4/4,'b')
grid on; xlabel('x'); ylabel('y')
legend(‘y = log(x)’, ‘Fourth-order approx at x=0.6‘)
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Fifth-order
10
y = log(x)
Fifth-order approx at x=0.6
8
6
y
4
2
0
-2
-4
0
0.2
0.4
0.6
0.8
1
x
1.2
1.4
1.6
1.8
2
x = linspace(0.05,2,50);
plot(x,log(x),'r',x, log(0.6)+(x-0.6)/0.6-(x-0.6).^2/0.6^2/2+(x-0.6).^3/0.6^3/3-(x-0.6).^4/0.6^4/4+(x-0.6).^5/0.6^5/5,'b')
grid on; xlabel('x'); ylabel('y')
legend(‘y = log(x)’, ‘Fifth-order approx at x=0.6‘)
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Taylor series
• Local approximation by power series.
• Try to approximate a function f(x) near x0, by
a0 + a1(x – x0) + a2(x – x0)2 + a3(x – x0)3 + a4(x – x0)4 + …
• x0 is called the centre.
• When x0 = 0, it is called Maclaurin series.
a0 + a1x + a2 x2 + a3 x3 + a4x4 + a5x5 + a6x6 + …
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Taylor series and Maclaurin series
Brook Taylor
English mathematician
1685—1731
Colin Maclaurin
Scottish mathematician
1698—1746
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Geometric series
Examples
Exponential function
Sine function
Cosine function
More examples at http://en.wikipedia.org/wiki/Maclaurin_series
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How to obtain the coefficients
• Match the derivatives at x =x0
• Set x = x0 in f(x) = a0+a1(x – x0)+a2(x – x0)2
+a3(x – x0)3+...
a0= f(x0)
• Set x = x0 in f’(x) = a1+2a2(x – x0) +3a3(x – x0)2+…
 a1= f’(x0)
• Set x = x0 in f’’(x) = 2a2+6a3(x – x0)
+12a4(x – x0)2+…
a2= f’’(x0)/2
– In general, we have ak= f(k)(x0) / k!
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Example f(x) = log(x), x0=0.6
• First-order approx.
log(0.6)+(x – 0.6)/0.6
• Second-order approx.
log(0.6)+(x – 0.6)/0.6 – (x – 0.6)2/(2· 0.62)
• Third-order approx.
log(0.6)+(x–0.6)/0.6 – (x–0.6)2/(2· 0.62)
+(x–0.6)3/(3· 0.63)
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Example: Geometric series
• Maclaurin series
1/(1– x) = 1+x+x2+x3+x4+x5+x6+…
• Equality holds when |x| < 1
• If we carelessly substitute x=1.1, then L.H.S. of
1/(1– x) = 1+x+x2+x3+x4+x5+x6+…
is equal to -10, but R.H.S. is not well-defined.
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Radius of convergence for GS
• For the geometric series 1+z+z2+z3+… , it
converges if |z| < 1, but diverges when |z| >
1.
• We say that the radius of convergence is 1.
• 1+z+z2+z3+… converges inside the unit disc,
and diverges outside.
complex plane
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Convergence of Maclaurin series in
general
• If the power series f(x) converges at a point x0,
then it converges for all x such that |x| < |x0|
Im
in the complex plane.
Re
x0
Proof by comparison test
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Convergence of Taylor series in
general
• If the power series f(x) converges at a point x0,
then it converges for all x such that
|x – c| < |x0 – c| in the complex plane.
Im
R
c
Re
x0
Proof by comparison test also
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Region of convergence
• The region of convergence of a Taylor series with
center c is the smallest circle with center c, which
contains all the points at which f(x) converges.
• The radius of the region of convergence is called
the radius of convergence of this Taylor series.
Im
diverge
R
c
Re
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Examples
•
: radius of convergence = 1. It converges
at the point z= –1, but diverges for all |z|>1.
• exp(z): radius of convergence is , because it
converges everywhere.
•
: radius of convergence is 0, because
it diverges everywhere except z=0.
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Behavior on the circle of
convergence
• On the circle of convergence |z-c| = R, a Taylor
series may or may not converges.
• All three series
 zn,  zn/n, and  zn/n2
Have the same radius of
R
convergence R=1.
But  zn diverges everywhere on |z|=1,
 zn /n diverges at z= 1 and converges at z=– 1 ,
 zn/n2 converges everywhere on |z|=1.
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Summary
• Power series is useful in calculating special
functions, such as exp(x), sin(x), cos(x), Bessel
functions, etc.
• The evaluation of Taylor series is limited to the
points inside a circle called the region of
convergence.
• We can determine the radius of convergence
by root test, ratio test, etc.
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