1.
Functions of a Real Variable
6. I njinite Series
I. A divergent series whose general term approaches zero.
To be specific we shall indicate a rearrangement of the series
{Sn} of partial sums has the
closed interval la, b) as its set of limit points. We start with the single
term 1, and then attach negative terms:
L( -l)n+I/n such that the sequence
The harmonic series I)/n.
La,.
2. A convergent series
and a divergent series
that a"
b n , n = I, 2, '" .
Let
an == 0 and b" ==
-1/n, n
=
Lan
n
such
1 - ! - ! - ! - ! - ...
1, 2, '" .
3. A convergent series
and a divergent series
that I a" I ~ I b" I, n = I, 2, '" .
Let
series
Lb
Lbn such
La"
be the conditionally convergent alternating harmonic
L( -1)n+l/n, and let Lb.. be the divergent harmonic senes
1-
Ll/n.
Lb
divergent series La,., and if Lb" converges it is dominated by the
convergent series Lan' The domination inequalities can all be made
strict by means of factors ! and 2.
This simple result can be framed as follows: There exists no positive
series that can serve simultaneously as a comparison test series for
convergence and as a comparison test series for divergence. (Cf. Example
I
1
2j
1
1
1
+ :3 + "5 + ... + 2k + 1
La,.
19, below.)
5. A convergent series with a divergent rearrangement.
With any conditionally convergent series Lan, such as the alter-l),,+l/n, the terms can be rearranged
nating harmonic series
L(
in such a way that the new series is convergent to any given sum,
or is divergent. Divergent rearrangements can be found SO that the
sequence (8,.j of partial sums has the limit + 00, the limit - 00, or
no limit at all. In fact, the sequence (snl can be determined in such
a way that its set of limits points is an arbitrary given closed interval,
bounded or not (cf. Example 2, Chapter 5). The underlying reason
that this is possible is that the series of positive terms of
and
the $eries of negative terms of
are both divergent.
La,.
1
2 - 4" - .•• -
until the Sum first is greater than b . Continuing with this idea we
adjoin negative terms until the sum first is less than a ; then positive
terms until the sum first exceeds b; then negative terms, etc., ad
infinitum. Since the absolute value l/n of the general term (-I)n+l/n
approaches zero, it follows that every number of the closed interval
la, b] is approached arbitrarily closely by partial sums Sn of the rearranged series, for arbitrarily large n. Furthermore, for no number
outside the interval [a, b] is this true.
In the procedure just described, if the partial sums are permitted
to go just above 1, then just below -2, then just above 3, then just
below -4, etc., the sequence of partial sums of the rearranged series
has the entire real number system as its set of limit points.
W. Sierpinski (cf. [43]) has shown that if
is a conditionally
convergent series with sum 8, and if s' < s, then for some rearrangement involving the positive terms only (leaving the negative terms
in their original positions) the rearranged series has sum s'. A similar
remark applies to numbers s" > 8 and rearrangements involving only
negative terms. This is clearly an extension of the celebrated "Riemann derangement theorem" (cf. [36], p. 232, Theorem III), illustrated in all its essentials by the discussion in Example 5.
In a different direction, there is an extension that reads: If La.. is
a conditionally convergent series of complex numbers then the sums
obtainable by all possible rearrangements that are either convergent
or divergent to 00 constitute a set that is either a single line in the
Complex plane (including the point at infinity) or the complex plane
in toto (including the point at infinity). Furthermore, if
is a
4. For an arbitrary given positive series, either a dominated
divergent series or a dominating convergent series.
A nonnegative series
is said to dominate a series
n iff
an ~ I b", I for n = 1, 2, .... If the given positive series is Lb",
let an
b" for n = 1, 2, '" . Then if Lbn diverges it dominates the
La,.
-.!.
2
4
6
8
2j
until the sum first is less than a. Then we add on unused positive
terms:
La,.
LV..
54
55
:·
MI
i
·r.' . .
,
I.
Fl£nctio1l.8 of a Real Variable
6. Infinite Series
c()nditionally convergent series of vectors in a finite-dimensional space,
then the sums obtainable by all possible rearrangements constitute
a sei; that is some linear variety in the space (cf. [47]).
8. A divergent series whose general term approaches zero and
which, with a suitable intrQduction of parentheses, becomes
convergent to an arbitrary sum.
Introduction of parentheses in an infinite series meanS grouping of
consecutive finite sequences of terms (each such finite sequence consisting of at least one term) to produce a new series, whose sequence
of partial sums is therefore a subsequence of the sequence of partial
sums of the original series. For example, one way of introducing
parentheses in the alternating harmonic series gives the series
La,.
fl. For an arhitrary conditionally convergent series
and
arbitrary real number x , a sequence {En! , where I En I = 1
for 11 = Ii 2, ... , such that
a,. = x.
Th.e procedure here is similar to that employed in Example 5.
Since LI a,,! =
<X) , we may attach factors Cn of absolute value 1
in such a fashion that C1 a1
Cn a" = I a1 I
I a" I > x.
Let r/l be the least value of n that ensures this inequality. We then
p:rovide factors cn , of absolute value 1, for the next terms in order to
obtain (for the least possible n2):
all
Len
+
el (tL
+ ... + C"2 a,,2
+ .,. +
= I al I +
+ ... +
(1-
1
= 1-2
... + J a,.1 1
I
an1 +I
1 -
.,. -
I
a.. 2 1 <
~) + G- i) + .. ,
1
+ 3·4 + ... + (2n -
1
1) ·2n
+ ....
Any series derived from a convergent series by means of introduction
of parentheses is convergent, and has a sum equal to that of the given
series.
The final rearranged series described under Example 5 has the
stated property since, for an arbitrary real number, a suitable introduction of parentheses gives a convergent series whose sum is the
given number.
x.
If this process is repeated, with partial sums alternately greater than
a" ~ 0
x and Jess than x, a series Lcn a" is obtained which, since
as n -+ + <fJ 1 must converge to x.
7. Divergent series satisfying any two of the three conditions
01 the standard alternating series theorem.
The alternating series theorem referred to states that the series
c,., where I Cn I = 1 and en > 0, n = 1, 2, '" ,converges
provided
n = 1,2, ... ,
(i) En = (-1)"+1,
n = 1, 2, .•. ,
(lit", c,. + L <
= en,
(iii) lir:n.n_+:o en = O.
No i;wo of these three conditions by themselves imply convergence;
that js, no one can be omitted. The following three examples demonstrate this fact:
.
(i): Let c"
1, en = lin, n = 1, 2, ..•. Alternatively, for an
example that is, after a fashion, an "alternating series" let {c"l be
th4:l ooquence repeating in triplets: 1,1, -1, 1,1, -1, ....
(ii): Let c" 0= lin if n is odd, and let en 0= ljn 2 if n is even.
1,
(iii): Let en == (n
1)ln (or, more simply, let en 0= 1), n
2, •..•
9. For a given positive sequence {b,,} with limit inferior zero,
La,.
Lc!'.,
a positive divergent series
whose general term approaches
zero and such that limn-+oo an Ibn = O.
Choose a subsequence bnl> b"u "', bnk' ••• of nonconsecutive
tenus of {b"l such that limk-+oo b"k = 0, and let a"k == b;k for k =
1, 2, .... For every other value of n : n = ml, m2, m3, ... ,m;, ... ,
let ami 0= Iii. Then a" ---? 0 as n ~
<x>, La.. diverges, and a"klb"k =
bn• ~ 0 as k ~ <fJ.
This example shows (in particular) that no matter how rapidly a
positive sequence {b,,! may converge to zero, there is a positive
sequence {a..l that converges to zero slowly enough to ensure divergence of the series La", and yet has a subsequence converging to
zero more rapidly than the corresponding subsequence of {bnl.
+
+
+
10. For a given positive sequence {b n } with limit inferior zero,
a positive convergent series
such that lim,.-+oo a n lbll = + <x).
La,.
56
57
~ ..
I.
Functions of a Real Variable
8.
preceding, let {i,,1 and {jkl be strictly increas~g sequences of positive
integers chosen so that the related propertIes hold as follows: We
choose i l and i l so that the related properties hold for T 1 ; then
i2 and h so that the related properties hold for both Tl and T 2 ; etc.
Let the sequence {anJ be defined as in Example 21. For any fixed m
the sequence {anI is transformed into a sequence {b:n n } and, since the
numbers i" and jk for k > m constitute sequences valId for the counterexample technique applied to T m, it. follows that limn-+co bmn does
not exist for any m.
successive term-by-term differentiation (which therefore all converge uniformly), is an infinitely differentiable function. Its Maclaurin
series has only terms of even degree, and the absolute value of the
term of degree 2k is
+co 2k
,,",xe
n 410
~o (2k)!
-1J,
I
is infinitely differentiable, all of its derivatives at x = 0 being equal·
to 0 (cf. Example 10, Chapter 3). Therefore its Maclaurin series
+co l")(o) "
2:--x
,,_0 nl
+co
=2:0
,,-0
converges for all x to the function that is identically zero, and hence
represents (converges to) the given function f only at the single point
x = O.
24,. A function whose Maclaurin series converges at only one
point.
A function with this property is described in [10], p. 153. The
function
+co
f(x) = 2: e-n cos n2x,
(2)2k
nx
> 2f
-n
e
for every n = 0, 1, 2, ... , and in particular for n = 2k. For this
value of n and, in terms of any given nonzero x, with k any integer
greater than e/2x, we have
22. A power series convergent at only one point. (Cf. Example
24.)
The series 2:!:'o n! x" converges for x = 0 and diverges for x 7"" O.
23. A function whose Maclaurin series converges everywhere
but represents the function at only one point.
The function
e-llx2 if x 7"" 0,
f(x) ==
if x
0
o
Infinite Series
(~;)2k e-1J, =
e: )2k
x
> 1.
This means that for any nonzero x the Maclaurin series for f diverges.
The series 2:t:o n! xn was shown in Example 22 to be convergent
at only one point, x = O. It is natural to ask whether this series is
the Maclaurin series for some function f(x) , since an affirmative
answer would provide another example of a function of the type
described in the present instance. We shall now show that it is indeed possible to produce an infinitely differentiable function f(x)
having the series given above as its Maclaurin series. To do this, let
f",o(x) be defined as follows: For n = 1 2 ... let
"
_f(n ¢"o(x) = .
l
1)1)2 if 0 ~ I x
0
if
Ix I ~
,
I~
2- n/(n!)2
2-n+1/(n!)2
where, by means of the type of "bridging functions" constructed for
Example 12, Chapter 3, ¢nO(x) is made infinitely differentiable every2,3"", let
where. Let!I(x) == ¢10(X), and for n
¢nl(X) ==
lX ¢nO(t) dt,
¢n2(X) ==
lX ¢nl(t) dt,
n=O
because of the factors e-n present in all of the series obtained by
68
69
6.
l'u'rlCiians of a Real Variable
I.
TlllJS jn'('J;)
f~I"')(V) =
= CP,.,fI_2(X), f,."(x) = CP",n_3(X), .,. ,f,.(n-lJ(x) = CPnO(X) ,
CPnO'(X). For any x and 0 ~ k ~ n - 2, 1i.. (k)(X) I ~
(2-+l/n:)I x
1,,-k- 2/(n - k - 2)! since
I CPnl(X) I ~ 2-n+1ln2,
j CP,,2(X) I ~ [2- n+1ln2]·1 x I,
for n = 1, 2, ... ,
1~" + 2;" + ... +
~~
1
1:
Infinite Series
[J(X)]2 dx.
Since f(x) is Riemann-integrable so is [j(x)]2, and the right-hand side
of the preceding inequality is finite, whereas if a ~ ~ the left-hand
side is unbounded as n -7 + 00. (Contradiction.)
+.,
.
. "smnx
The senes L.,; ~l-~ also converges for every real number x. Let
n
"=2
The series L~:tfn(k)(X), for each Ie = 0, 1, 2, ... , converges
UJl.aarmly in every closed finite interval. Indeed, if 1 x I ~ K,
n
f(x)
==
+00
L
n=2
•
sm nx .
In n
If f(x) is Lebesgue-integrable, then the function
F(x)
SlLd uniform convergence follows from the Weierstrass M-test (cf.
[34), p. 445). Hence we see that
+00
f(x)
== Lf..(x)
n=l
E
{'
f(t) dt
o
.
is both periodic and absolutely continuous. Since f(x) is an odd function (f(x) = -f( -x)), we see that F(:r) is an even function (F(x)
F( -x) and thus the Fourier series for F(x) is of the form
+«>
L an cos nx,
is an infinitely differentiable function such that for k = 0, 1, 2, ... ,
n=O
+00
fOcJ(x) = Lfn(l,)(x).
where ao =
ft=l
7r
Fad ~ n ~ 1,f.. (1·)(0) = CPnO(H'+l)(O) = O. Forn ~ 1 andk = n - 1,
j,•.c~)\O) = <1>.. 0(0) = «n - 1)1)2. For 0 ~ k < n - 1, fn(k}(O) = O.
Th1lS the Maclaurin series for f(x) is L!:o n! x".
2
an
+«>
The SEl1'jes
L
n=1
•
:X' where 0 < a
n
SIn
~
i,
converges for every real
llU]tlDer :x, as can be seen (d. [34], p. 533) by an application of a
eooTergelLce test due to N. H. Abel (Norwegian, 1802-1829). However, this series cannot be the Fourier series of any Riemann-integrable function f(x) since, by Bessel's inequality (cf. [34J, p. 532),
?O
=
7r
F(x) d.1;, and for n
0
~
2,
..
F(x) cos nx dx
0
~ F(x)
.2;. A convergent trigonometric series that is not a Fourier
series.
We shall present two examples, one in case the in+,egration involved
'is that of Riemann, and one in case the integration is that of Lebesgue.
1
1
1 ..
7r
---1
?"
7r
0
sin nx
n
I" - 21" F' (x) sinnnx dx
0
.
7ro
f( x ) sm n:v dx _
_
~
n
1
--~.
n In n
~F'(x) exists and is equal to f(x) almost everywhere.) Since F(x)
of. bound~d variation, its Fourier series converges at every point,
and III partICular at x = 0, from which we infer that L~:2 an converges. But since an
-l/(n In n), and L~:2 (-l/(n In n) diverges, we have a contradiction to the assumption that f(x) is
Lebesgue-integrable.
:IS
71
1
7.
Chapter 7
Uniform Convergence
Uniform Convergence
If x == 0, b...
Vii ~ + 00 as n ~ + 00, We shall show that for any
x r6 0 the sequence Ibn} is unbounded, and hence diverges, by showing
that there are arbitrarily large values of n such that I cos nx I ~ !.
Indeed, for any positive integer m such that I cos mx I < !,
I cos 2mx I
I 2 cos2 mx - 1. I =
SO that there exists an
n
>
1 - 2 cos 2 mx
m such that I cos nx
> !,
I > !.
3. A nonuniform limit of bounded functions that is not
hounded.
Each function
. ( 1)
1
Illin
Intl'oduction
The examples of this chapter deal with uniform convergence - and
OOJlvc:ugence that is not uniform - of sequences of functions on certain
sets. The basic definitions and theorems will be assumed to be known
(cf. [:14], pp. 441-462, [36], pp. 270-292).
I. A sequence of everywhere discontinuous functions con·
".n"~~
uniformly to an everywhere continuous function.
f .. (x)
lin if x is rational,
{o
if x is irrational.
+
00 < x <
00.
Tills simple example serves to illustrate the following general
CLearly. lim"+-T .. f .. (x) = 0 uniformly for -
prmeiple: Uniform convergence preserves good behavior, not bad be"f!tL-lli(ir. This same principle will be illustrated repeatedly in future
8::t;JLmples.
2. A sequence of infinitely differentiable functions converging
uniformly to zero, the sequence of whose derivatives diverges
eTerywhel'e.
Ii J", (x) == (sin nx)/yn, then since If ... (x) I ~ llyn this sequence
OOJlvelges uniformly to O. To see that the sequence {in'(X)} converges
nowhere, let x be fixed and consider
b.. = f ..'(x) =
Vii cos nx.
f,.(x)
==
n0 'x
if 0
<x
~
1,
if x = 0
is bounded on the closed interval [0, 1], but the limit function f(x),
equal to 1/x if 0 < x ~ 1 and equal to 0 if x = 0, is unbounded
there.
Let it be noted that for this example to exist, the limit cannot be
uniform.
4. A nonuniform liDlit of continuous functions that is not
continuous.
A trivial example is given by
f .. (x) == {min
(1, nx)
if x
max (-1, nx) if x
~ 0,
< 0,
whose limit is the signum function (Example 3, Chapter 3), which is
discontinuous at x = O.
A more interesting example is given by use of the function f (cf.
Example 15, Chapter 2) defined:
x = !!. in lowest terms, where p and q are integers
q
and q
> O.
x is irrational.
77
t.
Functions of a Real Variable
7.
Uniform Convergence
For an arbitrary positive integer n, define f"Cx) as follows: According
to each pomt
ort:h.e form
(~ , ~), where 1 ~ q < n, 0 ~
p
~
q, in each interval
1
(~ - 2~2' ~) define
(~, ~ + 2n
J»(x) == min
jn esc:h interval of the form
2
(x -
~)) j
i
(~ , + 2~2) define
J,,(x) == max
(~, ~
- 2n2 (x -
~)) ;
and at every point x of [0, 1J at which J71(x) has not already been
defined, let fn(x) == lin. Outside [0, 1J in(x) is defined so as to be
periodic with period one. The graph of in(x), then, consists of an
jnfinite polygonal arc made up of segments that either lie along the
horiZiontalline y = lin or rise with slope ±2n2 to the isolated points
of the graph of i. (Cf. Fig. 2.) As n increases, these "spikes" sharpen,
and the base approaches the x axis. As a consequence, for each
::c E CR and n = 1, 2,
0
1
2
!
3
!!
§
4
1
4
i
3
!!
2
In (x) for n~5
Figure 2
6. A sequence of functions for which the limit of the integrals
&Dd
lim f,,(x) = f(x),
"->+00
is not equal to the integral of the limit.
Let
as dEuned above. Each function in is everywhere continuous, but
the limit function f is discontinuous on the dense set ~ of rational
numbers. (Cf. Example 24, Chapter 2.)
2n 2;-c
in (:/:)
5. A llonuniform limit of Riemann -integrable functions that
is not Riemann-integrable. (Cf. Example 33, Chapter 8.)
Ea.ch function gn, defined for Example 24, Chapter 2, when restl'icted to the closed interval [0, 1J is Riemann-integrable there,
:smee it is bounded there and has only a finite number of points of
dise()ntinuity. The sequence /gn} is an increasing sequence Cgn(x) :;;:;
!T.t-l(x) for each x and n
1, 2, ... ) converging to the function f
of Example 1, Chapter 4, that is equal to 1 on ~ n [0, 1] and equal
to () on [0, 1] \ Q.
711
1
1
3
4
n
-
if 0:;;:; x
- 2n2(X
2~i)
0
Then
1
if
1
<
2n =
if
1
-:;;:;
n
1
2n'
~
X
1
< ,
=
n
X ~
L
1
lim
n-++ oo
fll(X) d:t:
0
but
1
{}
1
1
1
lim j,,(x) dx
n~+oo
0 dx = O.
79
I.
7.
Functions of a Real Variahle
Another example is the sequence {fn(X)} where fn(x) - nxe-n""
Then f" converges uniformly to 0 on [0,
1
O~x~1.
3
2n x
fn(x)
-
n
2
O~x~
if
-
2n3(X -
2~)
1
2n'
1
if n
f,,(x)
Then
~
x
~
1
fn(X) dx =
0
lim ?!: =
,,-+'" 2
while
b
lim fn(x) dx =
+ 00
'
+
1-
!~fn (x)
=
!~'"
(1
{I
2 2
+
nx
n 2x2 )2
=
if x
0 if 0
Let j n (X )
-
n
+t
O.
0
0
if x
>
~
•
+ 00.
0,
< Ixl ~
~
fn(x)
=
9. A sequence {fn} converging uniformly to zero on [0,
S;"'
1 (X) dx
= lin if
n 2,
11. A sequence converging nonuniformly and possessing a
uniformly convergent subsequence.
On the real number system CR, let
1.
==
{
if n is odd.
l'
11. 1f
.
n 18 even.
The convergence to zero is nonuniform, but the convergence of the
SUbsequence {f2n(x)! = {l/2n} is uniform.
8. Convergence that is uniform on every closed subinterval
but not uniform on the total interval.
Let fn(x) == xn on the open interval (0, 1).
such that
2
2:
Ib 0 dx = O.
7. A sequence of functions for which the limit of the derivatives is not equal to the derivative of the limit.
If fn(x) == x/(l
n2x2) for -1 ~ x ~ 1 and n = 1,2, ... , then
rex) == limn-+",fn(x) exists and is equal to 0 for all x E [-1,1] (and
this convergence is uniform since the maximum and minimum values
off7l(x) on [-1,1] are ± 1/2n). The derivative of the limit is identically equal to O. However, the limit of the derivatives is
•
= n ~
{~/n ~~ ox>~ nx ~
10. A series that converges non uniformly and whose general
term approaches zero uniformly.
The series
xnln on the half-open interval [0, 1) has these
properties. Since the general term is dominated by lin on [0, 1) its
uniform convergence to zero there follows immediately. The convergence of the series follows from its domination by the series
xn, which converges on [0, 1). The nonuniformity of this convergence is a consequence of the fact that the partial sums are not
uniformly bounded (the harmonic series diverges; cf. [34], p. 447,
Exs. 31, 32).
0
n~+~
I
S;'" f,,(x) dx
==
2:!:1
b
l
~ 1.
1,
ill which case, for any b E (0, 1]
lim
n-+'"
fn(X) dx = 1
A more extreme case is given by
1
1
x <if -2n <
=
= n'
0
•
+ 00), but
+'"
A more extreme case is given by
Uniform Convergence
12.-Nonuniformly convergent sequences satisfying any three
of the four conditions of Dini's theorem.
Dini's theorem states that if {fn} is a sequence of functions defined
on a set A and converging on A to a function f, and if
(i) fn is continuous on A, n = 1, 2, ... ,
+ 00) but
x ~ n,
n.
81
80
~..