Mathematical Physics I
— Infinite Series —
Minchul Lee
Department of Applied Physics
Physics
Resonance: transmission of light through a thin film of a thickness d
ttotal = teikd t + tr 2 eik (3d) t + tr 4 eik (5d) t + · · ·
2 4
= t 2 eikd 1 + reikd + reikd + · · · =
Ttotal = |ttotal |2 =
t 2 eikd
1 − r 2 e2ikd
|t|4
|t|4
=
2
2ikd
2
4
|1 − r e |
1 + |r | − 2 Re[r 2 ei2kd ]
Here 1 = t 2 + r 2 . See what happens for kd = nπ.
2
(1a)
(1b)
Physics (cont.)
Power-series calculation: power-series (numerical) calculation of special functions
∞
X
x3
x5
x 2n+1
+
− ··· =
(−1)n
3!
5!
(2n + 1)!
n=0
∞
s
X
n+2s
(−1)
x
sin x = x −
Jn (x) =
s=0
s!(n + s)!
2
(2a)
(2b)
Perturbation theory in quantum mechanics: power-series expansion of Schrödinger
equation — most useful approximation method in quantum physics
f (λ) → a0 + a1 λ + a2 λ2 + · · ·
where λ is assumed to be very small, |λ| ≪ 1.
3
(3)
Fundamental Concepts
Outline
1
Fundamental Concepts
2
Convergence Test
3
Algebra of Series
4
Series of Functions
5
Taylor’s Expansion
6
Power Series
4
Fundamental Concepts
Problems of Summation of an Infinite Number of Terms
Sequence of infinite terms
a1 , a2 , a3 , . . . , an , . . .
(4)
Partial sum: summation of a finite number of terms — always well-defined
sn ≡
n
X
am
(5)
m=1
Convergence of partial sums: If the partial sums sn converge to a finite limit as
n → ∞,
lim sn = S
n→∞
the infinite series
P∞
n=1
an is said to be convergent and to have the value S.
5
(6)
Fundamental Concepts
Examples of Convergent vs. Divergent Series
1
Oscillatory series — divergent series: For
an = (−1)n+1 :
1, −1, 1, −1, · · ·
(7)
the partial sums oscillate between 0 and 1:
sn =
n
X
(
am = 1 + (−1) + 1 + (−1) + · · · =
m=1
0, even n
1, odd n
(8)
There is no convergence to a limit → divergent
2
Simple example of divergent series: For
an = n,
the partial sums simply diverge
sn =
n
X
am = 1 + 2 + · · · + n =
m=1
n(n + 1)
−
−−−−
→ ∞
n→∞
2
Whenever the sequence of partial sums diverges, the infinite series is said to
diverge.
6
(9)
Fundamental Concepts
Examples of Convergent vs. Divergent Series (cont.)
3
Geometric series — convergent or divergent series: For
an = ar n−1 :
a, ar , ar 2 , · · ·
(10)
the particle sums are
sn =
n
X
ar m−1 = a
m=1
1 − rn
1−r
(11)
Therefore,
∞
X
an = lim sn = lim a
n=1
n→∞
n→∞
1 − rn
1−r
(12)
For |r | < 1, the limit exists so that
∞
X
n=1
an =
a
1−r
the infinite series converges.
For |r | ≥ 1, the limit does not exist (limn→∞ an ̸= 0) → the series diverges.
7
(13)
Fundamental Concepts
Examples of Convergent vs. Divergent Series (cont.)
4
Harmonic series – divergent series: For
an =
1
:
n
1 1 1
1, , , , · · ·
2 3 4
(14)
the infinite sum is divergent:
∞
X
n=1
1
1
1
+ + + ···
2
3
4
1
1
1
1
=1+ +
+
+
+
2 3
4 5
1
1
1
1
+
+
+
≥1+ +
2
4
4
8
1
1
1
= 1 + + + + ··· = ∞
2
2
2
an = 1 +
8
1
1
1
+ +
+ ···
6
7
8
1
1
1
+ +
+ ···
8
8
8
(15)
Fundamental Concepts
Convergence Condition
Naive condition for convergence
lim an = 0
n→∞
(16)
(?)
a counter-example: harmonic series
Note that
an is convergent
→
lim an = 0
n→∞
(17)
but the inverse is not always true. limn→∞ an = 0 is a necessary but not sufficient
condition for the convergence.
(Strict) Convergence condition (Optional):
Condition for the existence of a limit S
For each ϵ > 0, there is a fixed N such that
|S − sn | < ϵ
for n > N.
(18)
Cauchy criterion
For each ϵ > 0, there is a fixed number N such that
|sj − si | < ϵ
for all i, j > N
(19)
How to tell whether a given series is convergent or divergent → convergence tests
9
Convergence Test
Outline
1
Fundamental Concepts
2
Convergence Test
3
Algebra of Series
4
Series of Functions
5
Taylor’s Expansion
6
Power Series
10
Convergence Test
Comparison Test for Positive Series (an ≥ 0)
Basic Idea
1
0 ≤ an ≤ un for all n, un – already-known convergent series
X
X
0≤
an ≤
un = S → an is convergent
n
2
(20)
n
0 ≤ vn ≤ an for all n, vn – already-known divergent series
X
X
∞=
vn ≤
an → an is divergent
n
n
Note that there is no most slowly converging convergent series and no most slowly
diverging divergent series. This means that all convergent tests may fail sometime.
11
(21)
Convergence Test
Comparison Test for Positive Series (an ≥ 0) (cont.)
Cauchy Root Test — comparison with geometric series r n
1
2
If (an )1/n ≤ r < 1 for all n > N, with r independent of n, then
If (an )1/n
P
n
an is convergent.
0 ≤ an ≤ r n < 1
P
≥ 1 for all n > N, then n an is divergent.
(22)
an ≥ 1
IS-1: example —
P∞
n=1
(23)
n/10n
D’Alembert or Cauchy Ratio Test — comparison with geometric series r n
1
2
If an+1 /an ≤ r < 1 for all n > N, and r is independent of n, then
P
If an+1 /an > 1 for all n > N, then n an is divergent.
lim
n→∞
IS-2: example 1 —
P
n
an+1
an


< 1, convergent
= r > 1, divergent


= 1, indeterminant
n/2n
IS-3: example 2 — harmonic series
12
P
n
an is convergent.
(24)
Convergence Test
Comparison Test for Positive Series (an ≥ 0) (cont.)
Cauchy or Maclaurin Integral Test
Let f (x) be a continuous, monotonic decreasing function satisfying f (n) = an .
Then,
Z ∞
Z ∞
∞
X
f (x)dx <
f (x)dx + a1
(25)
an <
1
n=1
1
P
Hence n an converges or diverges as the integral converges or diverges. The
lower bound needs not be 1, but instead any number:
Z ∞
Z ∞
∞
X
f (x)dx <
an <
f (x)dx + aN
(26)
N
n=N
IS-4: example — harmonic series
IS-5: example — Riemann zeta function
13
N
Convergence Test
Convergence Test for Alternating Series
P
n+1
Leibnitz criterion: Consider the series ∞
an with an > 0. If an is
n=1 (−1)
monotonic decreasing (for all n > N) and limn→∞ an = 0, then the series
converges.
IS-6: proof of Leibnitz criterion
IS-7: example —
n+1
/n
n=1 (−1)
P
Estimation of error for partial sum:
|S − sn | = |(−1)n+2 (an+1 − an+2 + an+3 − · · · )|
= |an+1 − (an+2 − an+3 ) − (an+4 − an+5 ) − · · · | < |an+1 |
{z
} |
{z
}
|
>0
>0
The error in cutting off an alternating series after n terms is less than an+1 .
14
(27)
Convergence Test
Convergence Test for Alternating Series (cont.)
P
Absolute convergence:
Given a series an (in which an may vary
P
P in sign), if n |an |
converges,
then n an is said to be absolutely convergent. If n an converges but
P
|a
|
diverges,
the convergence is called conditional. Note that
n
n
X
|an |
→
convergent
n
X
n
IS-8: example — alternating harmonic series
15
an
convergent
(28)
Algebra of Series
Outline
1
Fundamental Concepts
2
Convergence Test
3
Algebra of Series
4
Series of Functions
5
Taylor’s Expansion
6
Power Series
16
Algebra of Series
Rearrangement of the Order
Rearrangement of the Order of Absolutely Convergent Series
Absolutely convergent series may be handled according to the ordinary familiar rules of
algebra or arithmetic:
1
If an infinite series is absolutely convergent, the series sum is independent of the
order in which the terms are added.
2
The series may be multiplied with another absolutely convergent series. The limit
of the product will be the product of the individual series limits. The product series,
a double series, will also converge absolutely.
IS-9: What about conditionally convergent series? In short, by a suitable rearrangement of terms a conditionally
convergent series may be made to converge to any desired value or even to diverge.
17
Algebra of Series
Improvement of Convergence: Rational Approximation
In numerical calculations of the sum of convergent series, the rate of convergence
is also an important factor to be considered.
Rational approximation: The rate of convergence may be improved substantially
by multiplying the infinite series by a polynomial and adjusting the polynomial
coefficients to cancel the more slowly converging portions of the series.
IS-10: series
ln(1 + x) =
=
∞
X
(−1)n−1 x n
n
n=1
1
1+x
x+
∞
X
n=2
18
(−1 < x ≤ 1)
(−1)n n
x
n(n − 1)
!
(29)
Algebra of Series
Rearrangement of Double Series (Optional)
As long as the double series an,m is absolute convergent,
∞ X
∞
X
an,m
(30)
m=0 n=0
rearrangement 1: n = q ≥ 0, m = p − q ≥ 0
∞ X
∞
X
an,m =
m=0 n=0
p
∞ X
X
aq,p−q
(31)
as,r −2s
(32)
p=0 q=0
rearrangement 2: n = s ≥ 0, m = r − 2s ≥ 0
∞ X
∞
X
an,m =
m=0 n=0
/2]
∞ [r
X
X
r =0 s=0
where [r /2] = r /2 for r even, (r − 1)/2 for r odd.
19
Series of Functions
Outline
1
Fundamental Concepts
2
Convergence Test
3
Algebra of Series
4
Series of Functions
5
Taylor’s Expansion
6
Power Series
20
Series of Functions
Series of Functions
Each term an may be a function of some variable such as x
an → an (x)
(33)
The partial sums become functions of the variable x
sn (x) = a1 (x) + a2 (x) + · · · + an (x)
(34)
Limit of the partial sums
∞
X
n=1
an (x) = lim sn (x) = S(x)
n→∞
21
(35)
Series of Functions
Uniform Convergence
Uniform Convergence
If for any small ϵ > 0 there exists a number N, independent of x in the interval [a, b]
such that
|S(x) − sn (x)| < ϵ,
for all n ≥ N,
the series is said to be uniformly convergent in the interval [a, b].
IS-11: example —
∞
X
n=1
x
((n − 1)x + 1)(nx + 1)
22
(36)
Series of Functions
Uniform Convergence (cont.)
Properties of uniformly convergent series
1
If the individual terms an (x) are continuous, the series sum
f (x) =
∞
X
an (x)
(37)
n=1
is also continuous.
2
If the individual terms an (x) are continuous, the series may be integrated term by term.
The sum of the integrals is equal to the integral of the sum:
Z b
∞ Z b
X
f (x)dx =
an (x)dx
(38)
a
3
n=1
a
The derivatives of the series sum f (x) equals to the sum of the individual term
derivatives
∞
X d an (x)
d f (x)
=
dx
dx
n=1
provided the following conditions are satisfied:
an (x) and dan (x)/dx are continuous in [a, b]
P∞
n=1 dan (x)/dx is uniformly convergent in [a, b]
23
(39)
Series of Functions
Uniform Convergence Test (Optional)
Weierstrass M Test
P∞
If one can find a convergent series
P∞ of numbers n=1 Mn , in which Mn ≥ |an (x)| for all x
in the interval [a, b], the series n=1 an is uniformly convergent in [a, b].
IS-12: proof of Weierstrass M test
Since we have specified
absolute values in the statement of the Weierstrass M
P
test, the series ∞
a
(x)
is also absolutely convergent.
n
n=1
In general uniform convergence and absolute convergence are independent
properties. Neither implies the other. For example,
∞
X
n=1
(−1)n−1
xn
= ln(1 + x)
n
0≤x ≤1
converges uniformly but does not converge absolutely.
24
(40)
Taylor’s Expansion
Outline
1
Fundamental Concepts
2
Convergence Test
3
Algebra of Series
4
Series of Functions
5
Taylor’s Expansion
6
Power Series
25
Taylor’s Expansion
Taylor’s Expansion
Taylor’s expansion = expansion of a function into an infinite series (a finite series +
remainder term)
Exact expansion of a function f (x) having a continuous nth derivative in the interval
[a, b]:
f (x) = f (a) + (x − a)f ′ (a) +
=
(x − a)2 ′′
(x − a)n−1 (n−1)
f (a) + · · · +
f
(a) + Rn (x)
2!
(n − 1)!
n−1
X
(x − a)s (s)
f (a) + Rn (x)
s!
s=0
(41)
with the remainder term
Rn (x) =
(x − a)n (n)
f (ζ),
n!
where ζ is a real number with a ≤ ζ ≤ x.
IS-13: proof of Eq. (41)
26
(42)
Taylor’s Expansion
Taylor’s Expansion (cont.)
Taylor’s expansion: If the function f (x) is such that
lim Rn (x) = 0
n→∞
(43)
the infinite series becomes Taylor’s series
f (x) =
∞
X
(x − a)n (n)
f (a)
n!
(44)
n=0
The infinite series converges because f (x) is well-defined function and the
expansion is exact.
IS-14: example 1 — f (x) = ex
IS-15: example 2 — f (x) = ln(1 + x)
Other form of Taylor’s expansion: With x → x + h and a → x
n
∞
∞
X
X
hn (n)
1
d
f (x + h) =
f (x) =
h
f (x)
n!
n!
dx
n=0
n=0
27
(45)
Taylor’s Expansion
Taylor’s Expansion (Optional)
Taylor’s expansion for more than one variable: For f = f (x, y )
∂f
∂f
+ hy
f (x + hx , y + hy ) = f (x, y ) + hx
∂x
∂y
1
∂2f
∂2f
∂2f
+
hx2 2 + 2hx hy
+ hy2 2
(46)
2!
∂x
∂x∂y
∂y
∂3f
1
∂3f
∂3f
∂3f
hx3 3 + 3hx2 hy 2
+
+ 3hx hy2
+ hy3 3
2
3!
∂x
∂x ∂y
∂x∂y
∂y
where all the derivatives are evaluated at the point (x, y ). In the vector form,
f (r + h) =
∞
X
1
(h · ∇)n f (r)
n!
n=0
28
(47)
Power Series
Outline
1
Fundamental Concepts
2
Convergence Test
3
Algebra of Series
4
Series of Functions
5
Taylor’s Expansion
6
Power Series
29
Power Series
Power Series
Power series: Representation of a (known or unknown) function in the form of
infinite series such as
f (x) =
∞
X
an x n = a0 + a1 x + a2 x 2 + · · · ,
(48)
n=0
where an are constants, independent of x.
Taylor’s expansion is an example of the power series.
Convergence ← Cauchy ratio test: If
lim
n→∞
an+1
= R −1 ,
an
the series converges for −R < x < R. Here R is called as the radius of
convergence. The convergence at end points x = ±R requires special attention.
IS-16: example — an = 1/n
IS-17: example — an = n!
P
Continuity: Since each of the term un (x) = an x n is continuous and f (x) = n an x n
converges uniformly for −R < S ≤ x ≤ S < R, f (x) must be a continuous function
in [−S, S].
30
Power Series
Power Series (cont.)
Differentiation: As long as an x n is uniformly convergent, the differentiated series is
a power series with continuous function and the same radius of convergence as
the original series.
∞
∞
n=1
n=1
X d an x n
X
df
=
=
nan x n−1
dx
dx
(49)
The power series may be differentiated or integrated as often as desired within the
interval of uniform convergence.
Uniqueness theorem: The power-series representation is unique. Suppose that
f (x) =
=
∞
X
n=0
∞
X
an x n ,
(−Ra < x < Ra )
(50)
bn x n ,
(−Rb < x < Rb )
(51)
n=0
with overlapping intervals of convergence (including the origin), then
an = bn
IS-18: proof of uniqueness theorem
31
(52)
Power Series
Power Series (cont.)
Importance of power-series representation
1
Development of solution of differential equations:
Jn (x) =
∞
X
s=0
2
(−1)s x n+2s
s!(n + s)! 2
Establishment of perturbation theory in quantum mechanics
IS-19: evaluate
lim
x→0
1 − cos x
x2
32