SUMMATION OF DIVERGENT INFINITE SERIES
BY ARITHMETIC, GEOMETRIC, AND HARMONIC
MEANS
by
FREDERICK HARRIS YOUNG
A THESIS
submitted to
OREGON STATE COLLEGE
In partial t'u1fi11nnt of
the requirements for the
degree of
MASTER OF ARTS
Juno j913
APPROVED:
Professor of Department of Mathematics
In Charge of Major
Head of Department of )4athematics
Chairman of School Graduate Comxuitte
Dean of Graduate Sohool
ACKNOWLEDGMENT
The writer wishes to express his deep gratitude
to Dr. C. L. Clark for the latter's skillful direction,
timely help, and abundant patience.
TABLE OF CONTENTS
Titlo
Chapter
I.
II
III.
INTROI5ITCTION.
1
,
.
.
s.,
.
* .
......
,
. , . . .
.1
TYPE S OF REQULA.R SUNMATI ON
À.
Cesàr'o Summation....
B.
aölder
C.
Nörilund Mean.
Suat ton.
.
. .
,
. . .
.
.
.
.
.
.
INTEGRAL I1EQJALIT)..
.
..
. .
.
.
. .
. .
.
.
.
. .
. s
s
.
.
. . . . .
s a
. *
... .3
. . . .
. a .
.
.
b
.
.
a
.9
EAN
HARMONIC-GEOMETRIC-ARITHMETIC
INEQUALITY.
IV.
.
. .
.10
I.
a a .
.
.
.
.
.
. .
.
.
.1)
V.
SUNATIONBYIIâRMOXUCNEÀL...............ILI
VI.
WEIGHTEDiiARMONICMEAN....................21
VII.
ZUMMATIONBYGEOMETRICKEAN...............22
VIII.
Ix.
WEIGHPEDGEOMETRICMEAL..................2k
X.
XIs
DIVERGENT INFINITE
PRODUCPS...............25
FURTHER EXAMPLES OF HARMONIC MEAN
COMPARISON OF PROPERTIES C? CESARO ui
HARMONIC MEANS. . . . . . . . . . . . . . . . . . . . . . . . .
.29
SUMMATION OF DIVERGENT INFINITE SERIES BY ARITHMETIC,
GEOMETRIC, AND HARMONIC MEANS
I.
INTROL1CTION
Prior to the time of Cauchy and Abel,
infinite
series vere used with no regard for the question of conIf a function could be expressed as an infinite
vergence.
sortes, well and good; the
eones represented the function.
This carefree attitude, which led to many discoveries
later validated by sounder means, was brought to a sudden
halt when Cauchy and Abel showed that the use of
series could lead to incorrect results.
sweep, divergent
divergent
Thus, with one
series were relegated to the limbo of
things tried and found wanting.
Not until ne8rly the end
of the nineteenth century did these treacherous traps for
the unwary begin to regain
Frobenius, in 1880,
an
aura of respectability.
extended the idea of convergency by
employing the limit of the
arithmetic
mean of the partial
sums of a series instead of the limit of the sum of the
first n ternis.
The work of Frobenius opened the way for a host
of workers.
vniue to
A way had been found to attach a unique
certain
nonconvergent series.
Further,
this
method, when applied to convergent series, yielded the
ordinary sum.
Within a decade Hölder, Cesro, and Borel
had successfully generalized the method of Frobenius,
2
and the early twentieth eentuiy
wiis
a period of intürthe
and fruitful activity in suimnation.
Certain terms 8tloUld be defined at this point.
A "regular" method of summation ta one that will yield
the ordinary value when
applied
to a convergent
series.
when applied to a nonconvergent
yields a unique result
series, the result is
known as a "conventional" sum.
Similar terms apply to
If,
in addition, this method
el;
o
infinite products.
The purpose
i.
of
this paper is es follows:
To define two new methods of summation, those of
the harmonic mean and of the geometric mean of
the partial sums of
a
series, and to show these
methods to be regular.
2.
To
define a
conventional product of an infinite
product as the geometric mean of the partial
products and to show this method is regular.
3.
Po compere these methods of summation with those
of Ces.ro, Hölder, and Nörlund.
3
TYPE S OF REGULAR BUMMAPI ON
II.
Cesro Summation
One of the moat useful methods of regular sum-
mation
that of Cesaro.
1.3
cO
n
(o)
Let
u.
Consider the series
.
U.
k=1
n
(i)
and,
=
in general, let
k=1
C
1(r))
(r)
(r)
Let
(r-1)
n
(r)
(o)
Let
rl(n
tri +
n
-
rfJT
s]
CJ
if for some r,
then,
;
has
00
a defluite limit, the series
u1,
1
is
snid to be sum-
k=1
mable (C,r).
Chapman and Knopp
S.
Cesro's method to the case
bility,
is not
r,
(r)
(5,
in which the order of summe-
necessarily integral.
+ n) denotes
n
(r+n
2...(r4-
(r - +- 11 jr +
and
r
)
(o?
=
s
-
(p
/)
(o)
s
=
fl(r
or
rCn +
8n1
p.26k), let
n
n.
+ i.)
2
For a proof of the
(7,
Then
(r)
, where
(
p.70) have extended
+ n + i)
i7Çr
"
(o)
8n-2
regularity of Cesro
1/r
+ n - 1\
n
cO
Ak=1
uk=lime
n
-
_!
-n:---
Jo
sumnii.tion
and
s
1f
Case
1.
A
n > O such that
Given any
(L
(m
il'
+
<
21'
+
I
there exists an
> O,
Hence,
if
n
-n
vfl
=
Thus,
!T
....
+
+
-
+
s
i
n
n
fr-/E,
Case 2.
to zero, and ve
lin
i
orltmç=O.
therefore, lin
O.
my
apply case
I to
- A)
(s,
n-'
P,
s,
of unity, mny converge to
!,
,
Sn'
w
LA(.Y
single lixaitin
o a
Howevei, the set of points
centioid of
2J
2'
happen that they do not converse
O;
'
A.
which rcpreent the numbers
is the
- A)
= lim(cr
tends
That is,
it.
n
If we consider the points
where
-
Then the sequence
A
n-'nk=1
point.
{21
+
nl
,
...
n'
2'
each wtth a masa
which thsn represents the
Cesàro sum.
For an illustration of Cesèro's method, lot us
consider the series I
(o)
-
I
+ I
-
i
+
(i)
=1
32n
...
.
Then
5
(i)
(o)
=0
(o)
=0
52n
(o)
8
2n + I
22n+1n
=1
Then C2
Thus,
(i)
(i)
(i)
n
I
C2
and 11m C
ri
+
2ñ+I
n-
2n+1=.
the original series is summable (C,i) to the value
1/2.
For an illustration of Cesro summation of a series
not swnmable (C,i), let us consider the series
u =
(o)
si
I(_l)kk.
k=1
Then
=_i
si
=_I
82n
=
=
'=-2
(i)
83
O
82n
=-2
(i)
(o)
=
O
=
2
.........................
(o)
(i)
$2n_1
82n-1
(i)
(o)
82n
=
S
-
I
-n(n + i)
(2)
(i)
(o)
(o)
S3
(2)
(i)
-
O
-n(n + i)
3
(2)
21(2n - i
f2n - I +
-
í-n'
t)
2-
i)
(n +
)
I7T
(2)
+n ;limC n-1=
n
-1
2n
(2)
212r1
-
(_)(n +
i)
i-
2n
(2)
limC 2n _-1.
(_i)kk is suinmable (C,2) to the
Thus, the series U =
value
k=1
-rn.
omehat different,
For another illustration,
sin la where x is not an even
consider the series U =
k=i
7.
multiple of
Then
n
s
sin la
=
n
cos x/2
-
k=1
2
-
cosjn+í/21x, and
-
sIr[x7
n
n
=
n
s
k=1
n cos x/2
n
2In
cos (k+1/2)x.
I
I1iE7
c7
Ic=1
n
k=1
n
+
- -
hm
sununable
O
(niJx
±
(c,i)
°
that
I cot x/2.
2iTh/2 =
if x is an even niultiple of r,
cos x/2
=
However,
U =
sin
iIn X7
2sTh x/2
fl
n-*oo
-
sin x
n cos x/2
flOE-.
Therefore,
-sin x + sin (n+i)x.
2sTn x/2
cos (k+1/2)x
Now,
O
+
...
and lin
noo
= O.
to the value I cot x/2 if
Therefore, U is
x
is not an even
7,
multiple of r 8nd to the value
O
otherwise.
roi
L!J
Holder Sumnmt ion
H6lder's method is siniilar to that of Cesàro, and
the two methods may be shown to be equivalent (3, p.85).
n
where
Consider the series
(i)
+
Let h
s
n
value of
r,
has
Uke
and,
in general, let
(r-1)
+h.
(r)
If 11m h
L.
k=1
+...+a,
(r-1)(r-1)
h1 4 h2+...
(r)
s
r
definite value for some integral
-
the sertes
2
Uk is said to be suinmable
(H,r).
Since Hldor's method is equivrlent to that of
Cesro,
it is
necessarily regular.
9
Nórlund Meen
Consider the sequence
consG1r1ts with C0
O,
fc1}
real or complex
oi'
c.
and let C
For any
k.o
ri
00
sortes
1
u,
s
where
and
k=o
k=o
cl-j:1
Uk,
CS
+
+
-
um
n
+ CrìSc
let
If li
S
"n
On
00
bae a definite value, the series
O,
-
n-
-
u
U
said to be
k=o
summable (N;c).
The method of Nrlund's Mean can be shown to be a
recular method of summation (k, p.37).
It is interesting to note that if all c
Nòrlund's iethod
Hölder for
te
:Ls
identical with those of
first order,
are i,
Cesro and
III.
HARMONIC-GEOMEPRIC-ARITBJIETIC MEAN INEQUALITY
inequality
A fundamentsl
which we shall find of
great use is the hrmonic-geometric-arithmetic mean in-
equality. The
(i, p.)25).
proof presented here is due to Cauchy
First, ve shall prove that the geometric mean of
several non-negative numbers A,B,C,D,... is always less
than their arithmetic mean.
Let n be the number of letters A,B,C,D,...
is
.
It
sufficient to prove that in general
A+B+C+D+...
BCD.Tt
(i)
n
or, what amounts to the same thing, that
n
-/A+B+C+D+
ABCD...
(2)
n
= (---
-
In the f trat place, for n = 2,
+ B2
___
-
A
-
2
- (
2)
(
) =
and one concludes, by taking successively n =
1
and finally n =
2m,
J
n =
k
J
'
________
________
¿
-
,
that
-
ABCflEFGH =
2
A + B
B
¿
_//
(A+B+c+D+E+F+a+H8,
_-
's___
__-)
n
-
--
+ ...)
ABCD... - (A + B +
- C + D
2m -_-- -J
term of the
In the second case, if n is not
(3)
,
'
8,
geometric progl'e8Siofl 2,l,8,16,...,
let us designate by
a term of this progression larger than n,
and we shell
let
K
A + B + C + D +
n
Then,
returning to formula
(3)
and suppOsing in the first
member of this formula that the last
2m
-
n factors are
equal to K, we twve
LJA + B
ABCD...K2
+C
+
J
L
or,
in other terme,
ABCD...K'
Then, multiplying both members by K'
.ABCD...
(A+B
K
,
we have
C+D
which was to be proved.
Nov, denote by A(a), O(a), and H(e) the arithmetic,
geometric, and harmonie means of a finite number (n) of
nonnegative numbers.
lin A(a)
uni
Also, for all a
and i/a(e) = 0(1/a).
Thus, 1/O(a)
1/il(e).
-
lin Hut)
exist.
O(a),
)
O,
,
we have
if the8e limita exist.
'e know that A(1/a) = 1/H(a),
Therefore,
i/o()
1/H(s), or H(a)
0(1/a)
lin G(a)
li
n-
n-*oo
i(a),
A(1/a) =
O(a), and
lin O(a), if these limits cxist.
lin li(a)
n-
Then, as n-*
Therefore,
if the limits
i'.
If the 1imit
of A(a),
G(), and H(a)
do not exist,
then the inequality holds true for the upper limite,
least upper bounds, etc. of the functions.
t)
IIEGRAL
IV.
I1EQ.UAIJITY
An interesting extension of the preceding inlet us define,
equality is the following:
terval O
<
A(f)
over the in-
ç X2,
f(x)dx,
J
x2_xix1
i
X
-(2
H(f)
2
-
log f (x ) dx
./
G(f)=eX21
X1
where f(x) has a positive lower
,
dx
Ixl
bound.
Á(f).
G(f)
Then H(f)
The general proof of this theoreni has been given by
Hardy, Littlewood, &ad Póly
inequality H(f)
(2,
p.135).
However, the
A(f) may be proved by Schwartz's
inequality.
The inequality is eviaentlj true ir
I
I
(x,.
C.
-x)2
-
ax
/
I f(x)dx, but by Schwartz's
d'-r
i
î
inequality,
rfX212
the latter member is not smaller thanf
dxj
L'j.
which is obviously (x2
-
x1)2.
3
Hence the inequality is
proved.
It can e&sily be shown that the general integral
inequality still holds a
x increases without bound.
1k
V.
SUMMATION BY HARMONIC MEANS
It has been shown that
the
summed by
suma.
an
uso of arithmetic
infinite series may
means of
be
the partial
We shall now show that this may aleo be done by
harmonic means as well.
the method of harmonic
Further,
means is regular; that is, when applied to a convergent
series it will yield the ordinary sum.
Consider the series
u
where again we
shell
k=1
n
let
s
S = 11m
U.
=
sn
If the series is convergent,
We must add the restriction that none of the
partial sums nor S be zero.
um
let
1,
i.
and it
has
Then,
been shown in section II that
n-
lims1+e2+...+s
S.
4CO
n
Let
p
The
p1
I and P
Then lin
1.
p=
P.
may now be considered as the partial sums of some
Thus
convergent series.
...
11m p1 + p2 +
n
__
n+ool+r
hm
1
s2
+ p
P,
or
=
..n
Hence, if a series is convergent in the ordinary sense,
the limit of the harmonic mean of the partial sums is
15
equal to the ordinary sui».
If,
That i», the method
however, the series to be tested is not ordinarily
convergent but the
limit
may be
We
series.
exists, the
conventional sum for the
limit of the harmonic
mean
considered to be a
shall denote b,r H the lii»
u
+..
i
s
It follows
when such exists.
the series
H.
is regular.
or
+
s
2
- 00
,
so F1180 will
k=1
The most interesting application of this method, then,
will
to series
be
whose partial sums oscillate between
infinite limits.
For an example, let
finite or
F1
Then
+
I - I
+
I - I
+
us consider the series
...
81=a
a + 1
83 =
s.
a + i
s
2 n-1
=a
=a+1.
2n-1
H21_
+T.
1
52n
Nov,
n
definition that if
from the
diverges to
u
I
i
a-+1
¡
I
2n -
___________
a
and
+1J
lia
H
2n-1
i
+
u + n -
¡
(2n - i)Ça)Sa +
2ia-+
2a(a
+
2a+1
i
i
i),
-1
Also,
2a(i.
2n__
n
H2
1
ñ+
swnblo
Thus, the given serLes is
to the value 2e(a
From
by
i).
+
this result
we
can see that
mean sum of a nonconvergent series
t, chancing
That
linear.
the harmonic mean
haraonic
not necessarily
a by an amount b does not
hmonic
necessarily change the
is
the
mean sum by
b.
Let us investigate the result of interpolating an
infinite
number of zeros in the
example.
Corwider the series
series of the previous
a+1+O-i+i+O-1+1+OThen
s=a+1
83n+1. =
=
a +
n
H3n+i =
Thus
lUn
Aleo
11
n+1+
a
!
H31
+2 = n
3n+
-i:---
and
1imH32
+
3a+I
+I+
- 3aIa +
The harmonic mean sum
(n
1)ja1a +
)na+n+a+1
2n
3a(a
-
i)
3a'a +
3n
2h
Now
And
j
i-
i)
i)
on
2
+ 2
3naÏ-
2ii-:i:--I
(a)[
n+
+ 1
¿a + 1
----:i:--i
i)
is
now
3a(a
a
+ i).
+-I--
Hence, the
(;
17
interpolation of an infinite number of zeros may affect
the
sum.
This is
also true of
Cesro summation. This
interpolation does not always affect the result, for an
examination of the series
a+1+O-1+O1+...
will show the harmonic mean. sum still 2a(a +
, as it
was without the zeros.
The typo of serios most frequently encountered in
summation
by means is that
number of
terna,
of terms, and O
H
n
.-.
=
in.
which, after a finite (k)
is recurrent
<
n
K
with a fixed period (a)
M,
n
+
and M finite.
ni
---+ .
..-
... +
(pk1+...+pk+a)=A. ThenHk+
For convenience, let (p1 +
+
and let
fl8
k+na
=
K+ii
k + na + d, where d is an integer such
and H
that
Then
I
d
a,
and D = p
«.
i-
k +
D,
can take on at most a finite (a) number of values.
then,
Then
liuiHk+na=lifllHk+na+da
From the last result it is evident that the
restriction must be added that A
O.
Under these con-
ditions, then, a series is summable by the harmonic mean
method.
It may easily be shown. that it
is al8o summable
under these conditions but not necessarily to the
same value, as we have seen in some of the previous
(c,i)
example s.
Suinmability by Cesro means does not necessarily imply suimaability by iarmonic means, nor is the
reverse true. For example, consider the sequence
N0E:
2'
1'
52n.
where the pair 82n - i'
1' '
or 1/10, 19/10. For n = 1, 2, ...
82n' 82n
is either 1,
(i)
i
(i)
LU +
,
and
=
either I or
(i)
Hence, liza
C
1
i.
2m +
1/10
I
=
_____________________ ,
Now, Hn
- I-:: I +-.;
I
+
n
'A
+1
I
82n-1
2
82n,
2n
where the
Then
pair I
82n-1
i..
î.
,
is either 1,
I
1
or 10, 10
82n
th aritbmetic mean of n pairs which are
2n
either
1,
1
or 10, 10/19.
The
aritbnietic
is 1, and that of IO, 10/19 is 100/19.
follows:
Construct si, 2'
' as
mean of 1,
I
let the first
19
two terms be
1.
1,
Then
1.
1
Now use the pair 1/10.
19/10 repeated enough times so that the reciprocal of
the harmonic mean for these terms exceed8 k. Next, use
enough of the pairs 1, 1 80 that the reciprocal of the
harmonic mean is less than 2. When this process is continued, a series is obtained for which the Cesro sum is
1, but the harmonic mean
oscillates,
never approaching a
limit. Obviously, a similar method could be used to
derive a series summable by harmonic neans but not by
Cesro a1eeris.
It is
nov
A slight change in notation is necessary.
monic means.
Let
possible to define higher orders of harn
:
-y
then
,
n
We have seen that
i1'
i
8
1
-p
;
.
-
Then
8.
let
n
-F...+
+
1?
Since
1H}
2i
S.
is a sequence
approaching the value
S,
In general,
H_____nrr.______
--
r n
.
r
-
and rHn
-
r -
-, S
We
may now
state
a
definition of order. If r is
20
the smallest integer such that urn
ki
u
is
H n exista, then
summable (rH) by harmonic means of order
r.
From the above, it is apparent that if a series is summable (rH)
If
k1
u
it is
sumniable
is convergent,
positive integer.
(r +
it is
1H) to the same value.
sumrnable (rl!) for r
21
TRE WEIGHTED HARMONIC MEAN
VI.
00
Cone ider the series
.
k=1
If the series is convergent,
n
u,
where s
urn s
u
=
Let
S.
O.
k=1
I
Sn
Then P = urn p n = urn I
flPco
I
.
Consider
n
Cn
,
=a1+cn_1P2+
n
C
=
,
and
um
O.
Then
en
IP2 +
+
n-Ç
k=1
+
where
s..
w=____ --cand
hm
I
= P
I
(by Norlund's Mean).
flc0
Thus,
him
= S,
and this method is
regular.
As
with
n-400
the Närlund mean, if all c
the ordinary harmonic mean.
= 1, this
method reduces to
2
SUMMATION BY GEOMETRIC MEAN
VII.
fl
2
Again consider
where
u1
s
k=1.
n
=
u
k=1
k
s
-
We may define a geometric conventional sum as
liai
(8152...8)hl'fl.
Let G
follows:
e)h/'1
Then
= G, the geometric conventional sum.
(2152..
Phis method is regular, for by section III we saw that
H
G
A,
and if
Uk is ordinari1
I
convergent to
S,
k=1.
H
A =
S.
Thus, G also equals 5.
exist when S does not.
However, G may
In that case G is a conventional
sum for the series.
There are certain restrictions that must be
Clearly, in order to
placed upon the geometric meen.
have a value different from zero, none of the partial
Also,
sums can be zero.
in order to avoid imaginary
we must insist that the product of the partial
results
sums be positive for
al].
n
where N is a finite
N,
integer.
For an illustration of this method, let us conaider our familiar series, a +
82n =
52n +
i
- i
+ i
-
i + ...
a + I
a
I
G2n = [a(a +
1)nj
1/2
I/2n
[a(a + i)J
O.
2
n+1
I
rn +
+
.
=
La
1
(a + I
)fl]2fl+'l
a
ri+L
ri
a
1)2flI
1/2
Then, urn
= '
fl-CK
Thui,
2n + i
the serles te suiuinable by the geometric mean
1/2
to the valuo
= [a(a + 1)7
lIa(a
+ l)J
VIII
TRE WEIGHTED GEOMETRIC MKAN
00
where
Once again consider the series
k=í
n
are greeter then zero.
and ali
=
Consider
k=1
i
s1)7vhere
as defined in section VII.
nd 11m in G
Therefore, G
+
.
= lin in
S if the
s
+ e1
ina
.
= in S, provided this exists.
is convergent to
&eries
k=1
S,
ere
Then
-
in
the e1 and C
and the method is regular.
X.
DIVERGENT
IN?INITE
PRODTJCT
In considering infinite series we have worked with
conventional sums, regular methods of summation that
yielded the ordinary sum when applied to convergent
series but that also gave rnrnerical values wten apolied
In a similar way we can
to certain nonconvergent series.
f
a method of evaluating infi1te products that is
md
regular; that is, Lb yields the ordinary product when
applied to a convergent infinite product but also may
be
extended to assign a numerical value to certain noncon-
vergent products.
Consider the product
Let
.
G
,
is convergent,
e
and let p
=
a1a2...a.
Then we have
in p1 + in p2 + ... +
-n
in G
If
TTa1
in
p
(in
thon the sequence
)
converges to lnîíwherelTis the ordtnry value of the
infinite product. Hence, by arithmetic mean summation,
in
G
- in iT,
does not
or urn G
exist, but um
=TF.
does
n-ì°o 0tl
However,
if um
exist, then urn
p
0ri
may be
co
considered to
Since
be a conventional product of
this
method is equivalent
IT
a,
to that of
Holder,
26
extend it easily to higher orders by taking the
limit of the root of the partial products of the partial
product s.
Let us apply this method to the infinite product
we may
(1)(2)(1/2)(2)(1/2)...
=1
p1
.
Now
=2
p2
...........
=2
2n
2n
= 1
+ ±
Obviously, the product does not converge.
flowever,
1/2 n
=
21/'2, and
(i)(2)(1)(2)...(2)]
I
02n +
Then, uni
= lu
=
[(I)(2)(1)(2)...(2)(1)]
a2
+
the conventional product.
21/'2.
n
=
Therefore, 21'2 ja
r
L
FURTHER EXAMPLE$ OF TILE &RMOtIC MEhN METHOD
Consider the sertes
In
tht8 SetieE,
first ten
I
I
U=1_
I
I
+ r
I
I
-
I
I
+ 7 -
+
I
+
.pplyirg, tc hernicnic ìieri method to the
approximation, we find that
The aun of the first ten terms is 0.61456.
terms to obtain
0.685.
&n
cf the infinite series is In 2 or 0.693. Hence,
the method of the haxinonic mean is in this case a better
approxiiwtion than is the partial sum.
Next, consider the series
The velue
U=2-5-2+7-5-2-i-7-5-2+7for which
=2
83
...
.
0
s.
2 fl
+
I
= 2
83n + 2
result in section V concerning
series with oscilleting partial sums are satisfied with
Here the conditions of the
a
= 3
and
A =
(-1/5
expect the result
+
lin
)-3X
1/2 - 1/3)
L
=
- 1/30.
)/(-I/30)
=
-90.
Hence, we
In order to
28
check
this',
we find
3n
-n + n
+ I =
+ 2
tht
-90,
-
n
+
-
33
__n +2
-n+1-1
,
11m
n-'°o
-90, anc
+ 2
= -90.
Thus, the series is summb1e (1H) to the value -90.
XI.
COMPARISON OF CESARO AND HARMONIC SUMMATION
Consider the series
(s/
M, M
vergerit,
let its
sum be
(i)
+
and
C
n
D
=
--I
W
-
+
k=1
where s
k=1
finite. If the series is
let
4H
con-
Let
denoted by S.
n , and
-
a,
U =
I.
3_.
n
-
We
may
i
3
J-
n
now state the following properties or characteristics of
Cesàro and harmonic summation:
()
is convergent, it is suxnmeble (c,i) and
to the value S.
2. If U ta eummï.ble (C,r), it is eummble (C,r + i),
and if it is aummable (rH) it is simmble r + 1H).
then both the
or - 00
3. If U diverges to +
arithmetic and harmonic means diverge to +aO or .00
k. IC U is suinniable (c,i) or (1H), it either converges
I. If
U
or oscillates
5.
If
U
fl
(i)
(i)
C1
between finite
=
is said to
11m Cr =
U
fl-3
6.
is said to
and
be bounded (c,i).
n1n=i;,
n-
'
or
infinite limits.
finite, thon
Similarly, if
I11n' 'nd$ftnite,
then
be bounded (1H).
The interpolation of a finite number of zeros does
30
not
'T.
or harmonic $um.
The interpolation of an infinite number of zeros
my
8.
chníe either the Cesro
chance both the Ces&ro and harmonic sums.
If U is summs.ble (C,r) and (.H), then rU is sum-
mable (C,r) and (rH) to r timos the former velues.
BIBLIOGRi.Pift
1.
OEvres coniplètes, series 2,
Cauchy, Augustin Loui3.
512p.
Parie, Gauthier-Villare, 1897.
vol. 3.
2.
Hardy, G. H., Littlewood, J. E., and P6lya, G. Inequalitiea. London, C8mbridge University press,
Jlkp.
193k.
3.
.
5.
Hobson, E. W. The theory of functions of a real
variable and the theory of Fourier's series,
London, Cambridge University press,
vol. 2.
1926.
78Op.
Summable sertes and convergence
Moore, Charles N.
factors. New York, American Mathematical
(American Mathematical
lO5p.
Society, 1938.
Society colloquium publications, vol. 22)
Aufgaben und lehrsitze aus
Berlin, Julius Springer, 1925.
Pólya, G. and Szegó', G.
der analysis.
JJ8p.
History and synopsis of the theory
Eugene,
of suznmable infinite processes 2:8.
175p.
1925.
publication,
University of Oregon
6.
Sznil, Lloyd L.
7.
New York,
Widder, David V. Advanced calculus.
k32p.
Prentice-Hall, Inc., 197.