Course 214
Section 2: Infinite Series
Second Semester 2008
David R. Wilkins
c David R. Wilkins 1989–2008
Copyright Contents
2 Infinite Series
2.1 The Comparison Test and Ratio Test .
2.2 Absolute Convergence . . . . . . . . .
2.3 The Cauchy Product of Infinite Series .
2.4 Uniform Convergence for Infinite Series
2.5 Power Series . . . . . . . . . . . . . . .
2.6 The Exponential Function . . . . . . .
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2
Infinite Series
An infinite series is the formal sum of the form a1 + a2 + a3 + · · ·, where each
+∞
P
number an is real or complex. Such a formal sum is also denoted by
an .
n=1
Sometimes it is appropriate to consider infinite series
+∞
P
an of the form
n=m
am + am+1 + am+2 + · · ·, where m ∈ Z. Clearly results for such sequences
may be deduced immediately from corresponding results in the case m = 1.
Definition An infinite series
+∞
P
an is said to converge to some complex
n=1
number s if and only if,
given any ε > 0, there exists some natural number N
m
X
an − s < ε for all natural numbers m satisfying m ≥ N . If
such that n=1
+∞
+∞
P
P
the infinite series
an converges to s then we write
an = s. An infinite
n=1
n=1
series is said to be divergent if it is not convergent.
For each natural number m, the mth partial sum sm of the infinite se+∞
+∞
P
P
ries
an is given by sm = a1 + a2 + · · · + am . Note that
an converges
n=1
n=1
to some complex number s if and only if sm → s as m → +∞. The following proposition therefore follows immediately on applying the results of
Proposition 1.2.
+∞
P
Proposition 2.1 Let
an and
n=1
+∞
P
λ
bn be convergent infinite series. Then
n=1
(an +bn ) is convergent, and
n=1
+∞
P
+∞
P
+∞
X
(an +bn ) =
n=1
+∞
X
n=1
an +
+∞
X
bn . Also
n=1
+∞
P
(λan ) =
n=1
an for any complex number λ.
n=1
If
+∞
P
an is convergent then an → 0 as n → +∞. Indeed
n=1
lim an = lim (sn − sn−1 ) = lim sn − lim sn−1 = s − s = 0,
n→+∞
where sm =
n→+∞
m
P
an and s =
n=1
n→+∞
+∞
P
n→+∞
an = limm→+∞ sm . However the condition
n=1
that an → 0 as n → +∞ is not in itself sufficient to ensure convergence. For
+∞
P
example, the series
1/n is divergent.
n=1
25
Proposition 2.2 Let a1 , a2 , a3 , a4 , . . . be an infinite sequence of real num+∞
P
an is convergent if and only if
bers. Suppose that an ≥ 0 for all n. Then
n=1
there exists some real number C such that a1 + a2 + · · · + an ≤ C for all n.
Proof The sequence s1 , s2 , s3 , . . . of partial sums of the series
+∞
P
an is non-
n=1
decreasing, since an ≥ 0 for all n. The result therefore is a consequence of
the fact that a non-decreasing sequence of real numbers is convergent if and
only if it is bounded above (see Theorem 1.3).
Example Let z be a complex number. The infinite series 1+z +z 2 +z 3 +· · ·
is referred to as the geometric series. This series is clearly divergent whenever
|z| ≥ 1, since z n does not converge to 0 as n → +∞. We claim that the
series converges to 1/(1 − z) whenever |z| < 1. Now
1 + z + z2 + . . . zm =
1 − z m+1
.
1−z
(To see this, multiply both sides of the equation by 1 − z.) Thus if |z| < 1
then
1
lim (1 + z + z 2 + . . . z m ) =
.
m→+∞
1−z
2.1
The Comparison Test and Ratio Test
Proposition 2.3 An infinite series
+∞
P
an of real or complex numbers is con-
n=1
vergent if and only if, given any ε > 0, there exists some natural number N
with the property that
|am + am+1 + · · · + am+k | < ε
for all m and k satisfying m ≥ N and k ≥ 0.
Proof The stated criterion is equivalent to the condition that the sequence
of partial sums of the series be a Cauchy sequence. The required result
thus follows immediately from Cauchy’s Criterion for convergence (Theorem 1.9).
Proposition 2.4 (Comparison Test) Suppose that 0 ≤ |an | ≤ bn for all n,
+∞
+∞
P
P
where an is complex, bn is real, and
bn is convergent. Then
an is
convergent.
n=1
26
n=1
Proof Let ε > 0 be given. Then there exists some natural number N such
that bm + bm+1 + · · · + bm+k < ε for all m and k satisfying m ≥ N and k ≥ 0.
But then
|am + am+1 + · · · + am+k | ≤ |am | + |am+1 | + · · · + |am+k |
≤ bm + bm+1 + · · · + bm+k < ε
when m ≥ N and k ≥ 0. Thus
+∞
P
an is convergent, by Proposition 2.3.
n=1
Let us apply the Comparison Test in the case when an and bn are non+∞
P
bn is convergent,
negative real numbers satisfying 0 ≤ an ≤ bn for all n. If
then so is
+∞
P
an . Thus if
n=1
+∞
P
an is divergent then so is
n=1
n=1
+∞
P
bn . These results
n=1
also follow directly from Proposition 2.2.
Example Comparison with the geometric series shows that the infinite series
+∞
P n
z /n is convergent whenever |z| < 1.
n=1
Proposition 2.5 (Ratio Test) Let a1 , a2 , a3 . . . be complex numbers. Sup+∞
P
an+1
exists and satisfies |r| < 1. Then
an is converpose that r = lim
n→+∞ an
n=1
gent.
Proof Choose ρ satisfying |r| < ρ < 1. Then there exists some natural
number N such that |an+1 /an | < ρ for all n ≥ N . Let
|a1 | |a2 | |a3 |
|aN |
,
,... N .
K = maximum
,
ρ ρ2 ρ3
ρ
Now |an+1 | ≤ ρ|an | whenever n ≥ N . Therefore |an | ≤ ρn−N |aN | ≤ Kρn
whenever n ≥ N . But the choice of K also ensures that |an | ≤ Kρn when n <
+∞
P
N . Moreover
Kρn converges, since ρ < 1. The desired result therefore
n=1
follows on applying the Comparison Test (Proposition 2.4).
Example Let z be a complex number. Then
+∞ n
X
z
converges for all values
n!
n=0
of z. For if an = z n /n! then an+1 /an = z/(n + 1), and hence an+1 /an → 0 as
n → +∞. The result therefore follows on applying the Ratio Test.
27
Let
+∞
P
an be an infinite series for which r = lim an+1 /an is well-defined.
n→+∞
n=1
The series clearly diverges if |r| > 1, since |an | increases without limit as
n → +∞. If however |r| = 1 then the Ratio Test is of no help in deciding
whether or not the series converges, and one must try other more sensitive
tests.
2.2
Absolute Convergence
P+∞
Definition An infinite
series
n=1 an is said to be absolutely convergent if
P+∞
the infinite series n=1 |an | is convergent. A convergent series which is not
absolutely convergent is said to be conditionally convergent.
An absolutely convergent infinite series is convergent, and the sum of
any two absolutely convergent series is itself absolutely convergent. These
results follow on applying the Comparison Test (Proposition 2.4). Moreover
the following criterion for absolute convergence follows directly from Proposition 2.3.
+∞
P
Proposition 2.6 An infinite series
an is absolutely convergent if and
n=1
only if, given any ε > 0, there exists some natural number N such that
|am | + |am+1 | + · · · + |am+k | < ε
for all m and k satisfying m ≥ N and k ≥ 0.
Many of the tests for convergence described above do in fact test for
absolute convergence; these include the Comparison Test and the Ratio Test.
2.3
The Cauchy Product of Infinite Series
The Cauchy product of two infinite series
+∞
P
an and
n=0
the series
+∞
P
+∞
P
bn is defined to be
n=0
cn , where
n=0
cn =
n
X
aj bn−j = a0 bn + a1 bn−1 + a2 bn−2 + · · · + an−1 b1 + an b0 .
j=0
The convergence of
+∞
P
n=0
an and
+∞
P
bn is not in itself sufficient to ensure the
n=0
convergence of the Cauchy product of these series. Convergence is however
+∞
+∞
P
P
bn are absolutely convergent.
assured provided that the series
an and
n=0
n=0
28
Theorem 2.7 The Cauchy product
+∞
P
cn of two absolutely convergent infi-
n=0
nite series
+∞
P
an and
n=0
+∞
P
bn is absolutely convergent, and
n=0
+∞
X
+∞
X
cn =
n=0
!
+∞
X
an
n=0
!
bn .
n=0
Proof For each non-negative integer m, let
Sm = {(j, k) ∈ Z × Z : 0 ≤ j ≤ m, 0 ≤ k ≤ m},
Tm = {(j, k) ∈ Z × Z : j ≥ 0, k ≥ 0, 0 ≤ j + k ≤ m}.
m m
m
P
P
P
P
P
Now
aj bk . Also
cn =
aj bk and
an
bn =
n=0
m
X
n=0
(j,k)∈Tm
X
|cn | ≤
n=0
n
P
since |cn | ≤
X
|aj ||bk | ≤
(j,k)∈Tm
n=0
(j,k)∈Sm
+∞
X
|aj ||bk | ≤
!
+∞
X
|an |
n=0
(j,k)∈Sm
|aj ||bn−j | and the infinite series
+∞
P
|bn | ,
n=0
+∞
P
an and
n=0
j=0
!
bn are absolutely
n=0
convergent. It follows from Proposition 2.2 that the Cauchy product
+∞
P
n=0
is absolutely convergent, and is thus convergent. Moreover
! m !
2m
m
X
X
X
bn cn −
an
n=0
n=0
n=0
X
= aj bk (j,k)∈T2m \Sm
X
X
≤
|aj bk | ≤
|aj bk |
(j,k)∈T2m \Sm
=
2m
X
!
|an |
n=0
(j,k)∈S2m \Sm
2m
X
!
|bn |
n=0
−
n=0
since Sm ⊂ T2m ⊂ S2m . But
! 2m
!
2m
X
X
lim
|an |
|bn |
=
m→+∞
m
X
n=0
29
|an |
n=0
+∞
X
!
|an |
n=0
=
!
lim
m→+∞
m
X
!
|bn | ,
n=0
+∞
X
!
|bn |
n=0
m
X
n=0
!
|an |
m
X
n=0
!
|bn | ,
cn
since the infinite series
+∞
P
an and
n=0
that
2m
X
lim
m→+∞
and hence
+∞
X
bn are absolutely convergent. It follows
n=0
m
X
cn −
n=0
!
an
n=0
cn = lim
m→+∞
n=0
+∞
P
2m
X
cn =
n=0
m
X
!!
bn
= 0,
n=0
+∞
X
n=0
!
an
+∞
X
!
bn ,
n=0
as required.
2.4
Uniform Convergence for Infinite Series
Let f1 , f2 , f2 , . . . be complex-valued functions defined over a subset D of C.
+∞
P
The infinite series series
fn (z) is said to converge uniformly on D to some
n=1
function s if, given any ε > 0, there exists some natural number
N (which
n
X
fm (z) < ε whenever
does not depend on the value of z) such that s(z) −
m=0
z ∈ D and n ≥ N .
+∞
P
Note that an infinite series
fn (z) of functions converges uniformly if
n=0
and only if the partial sums of this series converge uniformly. It follows
immediately from Theorem 1.20 that if the functions fn are continuous on
+∞
P
D, and if the series
fn (z) converges uniformly on D to some function,
n=0
then that function is also continuous on D.
Proposition 2.8 (The Weierstrass M -Test) Let D be a subset of C and let
f1 , f2 , f3 , . . . be a sequence of functions from D to C, let M1 , M2 , M3 , . . . be
non-negative real numbers satisfying |fn (z)| ≤ Mn for all natural numbers n
+∞
+∞
P
P
and z ∈ D. Suppose that
Mn converges. Then
fn (z) converges abson=1
n=1
lutely and uniformly on D.
Proof It follows immediately from the Comparison Test (Proposition 2.4)
+∞
P
that the series
fn (z) is absolutely convergent for all z ∈ D. We must
n=1
show that the convergence is uniform.
Let ε > 0 be given. Then there exists some natural number N such
+∞
P
P
1
that +∞
M
<
ε,
since
Mn converges. Now if m and k are integers
n
n=N
2
n=1
30
satisfying m ≥ N and k ≥ 0 then
m+k
+∞
m
m+k
m+k
X
X
X
X
X
Mn ≤
Mn < 21 ε
fn (z) ≤
fn (z) −
fn (z) = n=m+1
n=1
n=m+1
n=1
n=N
for any z ∈ D. On taking the limit as k → +∞, we see that
+∞
m
X
X
fn (z) −
fn (z) ≤ 21 ε < ε
n=1
n=1
for all z ∈ D and m ≥ N . However N has been chosen independently of z.
Thus the infinite series converges uniformly on D, as required.
2.5
Power Series
A power series is an infinite series of the form
+∞
P
an (z − z0 )n , where the
n=0
coefficients a0 , a1 , a2 , . . . are complex numbers.
Definition Let
+∞
P
an (z − z0 )n be a power series centred on some complex
n=0
number z0 . Suppose that the set of complex numbers z for which the power
series converges is bounded. Then the radius of convergence R0 of the power
series is defined to be the smallest non-negative real number with the property
that every complex number z for which the power series converges satisfies
|z − z0 | ≤ R0 . The circle {z ∈ C : |z − z0 | = R0 } is then referred to as
the circle of convergence of the power series. We set R0 = +∞ if the set of
complex numbers z for which the power series converges is unbounded.
Theorem 2.9 Let
+∞
P
an (z − z0 )n be a power series with radius of conver-
n=0
gence R0 , and let s(z) denote the sum of the power series at those complex
numbers z at which the series converges.
(i) If R0 = +∞ then s(z) is a continuous function of z defined over the
entire complex plane C.
(ii) If R0 < +∞ then s(z) is a continuous function of z defined over the
whole of the disk
{z ∈ C : |z − z0 | < R0 }
bounded by the circle of convergence of the power series.
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Proof Let z1 be any complex number satisfying |z1 − z0 | < R0 . Then we can
choose R such that |z1 −z0 | < R < R0 and R < +∞. Now it follows from the
definition of the radius of convergence that there exists some complex num+∞
P
ber w such that R < |w| < R0 and
an wn converges. Choose some positive
n=0
real number A with the property that |an wn | ≤ A for all n, and set ρ = R/|w|
n
and Mn = Aρ
. If |z − z0 | < R then |an (z − z0 )n | ≤ |an |Rn ≤ Aρn = Mn for
P+∞
all n. Also n=0 Mn converges to A/(1 − ρ). Thus we can apply the Weier+∞
P
strass M -Test (Proposition 2.8) to deduce that the power series
an (z−z0 )n
n=0
converges uniformly on the disk {z ∈ C : |z − z0 | < R} of radius R about z0 .
It then follows from Theorem 1.20 that the restriction of the function s to
this disk is continuous on the disk, and, in particular, is continuous around
z1 . We deduce that the function s is continuous throughout the complex
plane when R0 = +∞, and is continuous inside the circle of convergence
when R0 < +∞, as required.
A power series with finite radius of convergence will converge everywhere
within its circle of convergence, and will diverge everywhere outside this circle. However Theorem 2.9 provides no information concerning the behaviour
of the power series on the circle of convergence itself.
2.6
The Exponential Function
Definition The exponential function exp: C → C is defined for all complex
numbers z by the formula
exp(z) =
+∞ n
X
z
n=0
n!
.
A straightforward application of the Ratio Test shows that radius of convergence of the power series defining exp(z) is infinite, and it therefore follows
from Theorem 2.9 that the exponential function is continuous on C.
Lemma 2.10 The exponential function has the property that
exp(z + w) = exp(z) exp(w).
for all complex numbers z and w.
Proof Let z and w be complex numbers. The infinite series defining exp(z)
and exp(w) are absolutely convergent. It therefore follows from Theorem 2.7
32
that the value of exp(z) exp(w) is the sum of the Cauchy product of the
infinite series defining exp(z) and exp(w), and therefore
exp(z) exp(w) =
+∞
X
cn
n=0
where
cn =
n
X
zj
j=0
wn−j
j! (n − j)!
On applying the Binomial Theorem, we see that
n
n n!
1
1 X
1 X n j n−j
j n−j
z w
= (z + w)n ,
cn =
z w
=
n! j=0 j!(n − j)!
n! j=0 j
n!
and thus exp(z) exp(w) = exp(z + w), as required.
Let z be a complex number. Then exp(z) exp(−z) = exp(z − z) =
exp(0) = 1. It follows that exp(z) 6= 0 and exp(−z) = 1/ exp(z) for all
complex numbers z.
Let sin: C → C and cos: C → C be the functions defined as sums of power
series according to the formulae
sin z =
∞
X
(−1)n z 2n+1
n=0
(2n + 1)!
,
cos z =
∞
X
(−1)n z 2n
n=0
(2n)!
.
If z is real then the power series appearing on the right hand side of these
identities correspond to the Taylor expansions of the standard sine and cosine
functions. A routine application of the Ratio Test shows that these power series both have infinite radius of convergence, and therefore define continuous
functions defined over the whole complex plane. These continuous functions
therefore constitute a natural extension to the complex plane of the standard
sine and cosine functions of basic trigonometry.
On comparing power series, we see that
1
cos z = (exp(iz) + exp(−iz)),
2
cos z + i sin z = exp(iz),
sin z =
1
(exp(iz) − exp(−iz)),
2i
cos z − i sin z = exp(−iz)
for all complex numbers z. Thus if z = x + iy, where x and y are real
numbers, then
exp(z) = exp(x + iy) = exp(x) exp(iy) = ex (cos y + i sin y).
33
The exponential map exp: C → C \ {0} is surjective. For, given any
complex number w, there exist real numbers r and θ such that r > 0 and
w = r(cos θ + i sin θ). Moreover there exists a real number x such that
r = ex . (This real number x is the natural logarithm loge r of r.) Then
w = exp(x + iθ).
Lemma 2.11 Let z and w be complex numbers. Then exp(z) = exp(w) if
and only if w = z + 2πin for some integer n.
Proof If w = z + 2πin for some integer n then
exp(w) = exp(z) exp(2πin) = exp(z)(cos 2πn + i sin 2πn) = exp(z).
Conversely suppose that exp(w) = exp(z). Let w−z = u+iv, where u, v ∈ R.
Then
eu (cos v + i sin v) = exp(w − z) = exp(w) exp(z)−1 = 1.
Taking the modulus of both sides, we see that eu = 1, and thus u = 0. Also
cos v = 1 and sin v = 0, and therefore v = 2πn for some integer n. The result
follows.
Proposition 2.12 Let D0 = C \ {x ∈ R : x ≤ 0}. Then there exists a
unique continuous function log: D0 → C characterized by the properties that
log(1) = 0, | Im log(z)| < π and exp(log(z)) = z for all z ∈ D0 . This
function has the property that log(reiθ ) = loge r + iθ for all real numbers r
and θ satisfying r > 0 and −π < θ < π, where loge r denotes the natural
logarithm of the real number r.
Proof Given any complex number w belonging to the open set D0 there
exist unique real numbers r and θ for which r > 0, −π < θ < π and
reiθ = w. It follows that there is a well-defined function log: D0 → C such
that log(reiθ ) = loge r+iθ for all real numbers real numbers r and θ satisfying
r > 0 and −π < θ < π, where loge r, the natural logarithm of r, is the unique
real number t satisfying et = r. Moreover
exp(log(reiθ )) = exp(loge r) exp(iθ) = reiθ
for all real numbers r and θ with r > 0. It follows that exp(log(z)) = z for
all z ∈ D0 .
34
Let x and y be real numbers. Then
!


x

1

loge (x2 + y 2 ) + i arccos p
if y > 0,

2

2 + y2

x


!


x
if y < 0,
log(x + iy) = 21 loge (x2 + y 2 ) − i arccos p
2 + y2

x


!



y

2
2
1


if x > 0,
 2 loge (x + y ) + i arcsin p 2
x + y2
where arcsin: [−1, 1] → [−π/2, π/2] and arccos: [−1, 1] → [0, π] are the inverses of the restrictions of the sine and cosine functions to the intervals
[−π/2, π/2] and [0, π] respectively. Now the functions loge , arcsin and arccos
are continuous, since the inverse of any strictly increasing or strictly decreasing function defined over some interval in R is itself continuous. (see
Corollary 1.19). We conclude that the restrictions of log: D0 → C to each of
the open sets
{z ∈ C : Im z > 0},
{z ∈ C : Im z < 0},
{z ∈ C : Re z < 0}
is continuous, and the open set D0 is the union of these three open sets. It
follows that log: D0 → C is continuous throughout D0 . A straightforward
application of Lemma 2.11 then shows that this function log is the unique
function from D0 to C with the required properties.
The function log: D0 → C characterized by the properties set out in
the statement of Proposition 2.12 is referred to as the principal branch of
the logarithm function. It is impossible to define a continuous ‘logarithm’
function L: C \ {0} → C with the property that exp(L(z)) = z for all z ∈
C \ {0}. The best that one can achieve is to define continuous inverses of the
exponential map over appropriately chosen open subsets of C \ {0}. Such
functions are referred to as ‘branches of the logarithm function’.
Corollary 2.13 Let w be a non-zero complex number, and let
Dw,|w| = {z ∈ C : |z − w| < |w|}.
Then there exists a continuous function Fw : Dw,|w| → C with the property
that exp(Fw (z)) = z for all z ∈ Dw,|w| .
Proof Let D0 = C \ {x ∈ R : x ≤ 0}, let log: D0 → C be the principal
branch of the logarithm function, and let ζ be a complex number satisfying
exp ζ = w. Then z/w ∈ D0 for all z ∈ Dw,|w| . A function Fw : Dw,|w| → C
with the required properties may therefore be obtained on defining Fw (z) =
ζ + log(z/w) for all z ∈ Dw,|w| .
35