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

Chapter Titlo

I. INTROI5ITCTION. . 1 , . . s., . , . , . . . .1

II TYPE S OF REQULA.R SUNMATI ON

À. Cesàr'o Summation.... , * . ...... . ... .3

B. aölder Suat ton. . . . . . . . . . . . . . . . . . . . b

C. Nörilund Mean. . s . s s a . * . a . . . a .9

III. HARMONIC-GEOMETRIC-ARITHMETIC EAN INEQUALITY. . . . . . . . . . . . . .. . . I. .10

IV. INTEGRAL I1EQJALIT).. . . . . . . a a . . . . . . . . . . .1)

V. SUNATIONBYIIâRMOXUCNEÀL...............ILI

VI. WEIGHTEDiiARMONICMEAN....................21

VII. ZUMMATIONBYGEOMETRICKEAN...............22

VIII. WEIGHPEDGEOMETRICMEAL..................2k

Ix. DIVERGENT INFINITE PRODUCPS...............25

X. FURTHER EXAMPLES OF HARMONIC MEAN

XIs 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 con-

vergence. If a function could be expressed as an infinite

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 divergent

series could lead to incorrect results. Thus, with one

sweep, divergent 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 an aura of respectability.

Frobenius, in 1880, 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. A way had been found to attach a unique

vniue to 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

the ordinary value when applied

If, in addition, this method el;

when applied to a nonconvergent

known as a "conventional" sum.

infinite products.

ta one that will yield

to a convergent series. o yields a unique result

series, the result is

Similar terms apply to

The purpose of this paper is es follows:

i. 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

II. TYPE S OF REGULAR BUMMAPI ON

Cesro Summation

One of the moat useful methods of regular sum-

mation 1.3 that of Cesaro. cO (o) n

Consider the series u. Let . U. k=1

(i) n (o) (r) n (r-1) Let and, in general, let =

k=1

(r) (r) 1(r))

Let C rl(n - s] ; then, if for some r, CJ has n

tri + rfJT a defluite limit, the series

mable (C,r).

00

1 u1, is snid to be sum- k=1

S. Chapman and Knopp (5, p.70) have extended

Cesro's method to the case in which the order of summe-

bility, r, is not necessarily integral. Then

(r) (r) , where + n) denotes

(r+n n

(r + 11 jr + 2...(r4- n or fl(r + n + i) - - n. rCn + i7Çr 1f

( r ) (o? (p (o) /) + i.) (o) 1/r + n - 1\

and = s - 8n1 2 8n-2 " n Jo

For a proof of the regularity of Cesro sumnii.tion

(7, p.26k), let s = s and

cO A- uk=lime

k=1 n - -n:---

_!

Case 1. A (L Given any > O, there exists an

n > O such that (m + il' I + 21' < Hence, if

n

+ + .... + s i + {21 - vfl n - nl

= n n

Thus, !T fr-/E, orltmç=O. Case 2. A O. Then the sequence - tends

to zero, and ve my apply case I to it. That is,

n lin i (s, - A) = lim(cr - A) O;

n-'nk=1 n-'

therefore, lin A.

If we consider the points P, 2J ' which rcpreent the numbers s, 2' Sn'

w LA(.Y

happen that they do not converse o a single lixaitin

point. Howevei, the set of points , 2' n'

where is the centioid of , ... each wtth a masa

of unity, mny converge to !, which thsn represents the

Cesàro sum.

For an illustration of Cesèro's method, lot us

consider the series I - I + I - i + ... . Then

(o) (i) =1 32n

5

(o) (i) =0 22n+1n

(o)

52n =0

(o) 8 =1 2n + I

(i) (i) (i)

Then C2 n I C2 + ri and 11m C

2ñ+I n- 2n+1=.

Thus, 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 = I(_l)kk. Then k=1

(o) (i)

si =_i si

=_I

(o) (i) = = O

(o) (i) S3 '=-2 83 =-2

(o) (i) = 2 = O

.........................

(o) (i)

$2n_1 82n-1

(o) (i)

82n = S - O

(2)

82n I -n(n + i)

-

(2)

82n -n(n + i)

3 (2)

21(2n - i - t) í-n' (n + i) f2n - I + 2- I7T )

(2) n +n ;limC -1 n-1=

2n

(2)

i-

(_)(n + i) 212r1

-

2n

(2) limC _-1.

2n -

Thus, the series U = (_i)kk is suinmable (C,2) to the k=1

value -rn. For another illustration, omehat different,

consider the series U = sin la where x is not an even k=i

multiple of 7. Then n

s = sin la cos x/2 - cosjn+í/21x, and n

k=1 - 2 sIr[x7 - n n

= s n cos x/2 I cos (k+1/2)x. n

k=1 n 2In c7 I1iE7 Ic=1

n Now, cos (k+1/2)x -sin x + sin (n+i)x. Therefore,

k=1 2sTn x/2

n cos x/2 sin x - sin (niJx flOE-. + iIn X7 ° that n - - 2sTh x/2

hm cos x/2 I cot x/2. n-*oo fl = 2iTh/2 =

However, if x is an even niultiple of r,

U = O ± O + ... and lin = O. Therefore, U is noo

sununable (c,i) to the value I cot x/2 if 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

Consider the series where s L. Uke k=1

(i) Let h + s +...+a, and, in general, let

(r) (r-1)(r-1) (r-1) h1 4 h2+... +h.

(r) If 11m h has r definite value for some integral

n -

value of r, 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 fc1} oi' real or complex

consG1r1ts with C0 O, and let C c. For any k.o

00 ri sortes 1 u, where s Uk, and um n O, let

k=o k=o n- -

cl-j:1 CS + + + CrìSc S If li -

On "n 00

bae a definite value, the series - 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 are i,

Nòrlund's iethod :Ls identical with those of Cesro and

Hölder for te first order,

III. HARMONIC-GEOMEPRIC-ARITBJIETIC MEAN INEQUALITY

A fundamentsl inequality which we shall find of

great use is the hrmonic-geometric-arithmetic mean in-

equality. The proof presented here is due to Cauchy

(i, p.)25). 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,... . It

is sufficient to prove that in general

(i) BCD.Tt A+B+C+D+... n

or, what amounts to the same thing, that

(2) ABCD... -/A+B+C+D+ n

= (--- n -

In the f trat place, for n = 2,

+ B2 A - B A + B 2

- ___ - ( 2 ) = ( 2)

and one concludes, by taking successively n = , n = 8,

1 and finally n = 2m, that -

J ' J k

ABCflEFGH ________ ________ = - ¿ _//

¿

(A+B+c+D+E+F+a+H8, - 's___ _- - __-)

n

(3) ABCD... , (A + B + C + D + ...) - -- - 2m -_-- -J

In the second case, if n is not ' term of the

-

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

ABCD...K2 LJA + B +C + L J

or, in other terme,

ABCD...K'

Then, multiplying both members by K' , we have

.ABCD... K (A+B 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. Then, as n-* , we have

lin A(a) uni O(a), if the8e limita exist.

Also, for all a ) O, 'e know that A(1/a) = 1/H(a),

and i/a(e) = 0(1/a). Therefore, i/o() 0(1/a) A(1/a) =

1/il(e). Thus, 1/O(a) 1/H(s), or H(a) O(a), and

lin Hut) lin O(a), if these limits cxist. Therefore, - lin li(a) lin G(a) li i(a), if the limits n- n- n-*oo

exist.

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)

IV. IIEGRAL I1EQ.UAIJITY

An interesting extension of the preceding in-

equality is the following: let us define, over the in-

terval O < ç X2,

A(f) i J f(x)dx, x2_xix1 X -(2

./ log f (x ) dx G(f)=eX21 X1

H(f) 2

- , where f(x) has a positive lower

dx I xl

bound. Then H(f) G(f) Á(f).

The general proof of this theoreni has been given by

Hardy, Littlewood, &ad Póly (2, p.135). However, the

inequality H(f) A(f) may be proved by Schwartz's

inequality.

The inequality is eviaentlj true ir

I I

(x,. -x)2 / ax I f(x)dx, but by Schwartz's C. -

d'-r

î i rfX212 inequality, the latter member is not smaller thanf dxj

L'j. 3

which is obviously (x2 - x1)2. 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 an infinite series may be

summed by the uso of arithmetic means of the partial suma. We shall now show that this may aleo be done by

harmonic means as well. Further, the method of harmonic

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 = U. If the series is convergent, let

S = 11m s We must add the restriction that none of the n

partial sums nor S be zero. Then,

um i. 1, and it has been shown in section II that n-

lims1+e2+...+s S. 4CO

n

Let p I and P 1. Then lin p= P.

The p1 may now be considered as the partial sums of some

convergent series. Thus

11m p1 + p2 + ... + p P, or =

n

hm __ n n+ool+r ..

1 s2

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». That i», the method is regular.

If, however, the series to be tested is not ordinarily

convergent but the limit of the harmonic mean exists, the

limit may be considered to be a conventional sum for the

series. We shall denote b,r H the lii» u i +.. i

s s I 2 n

when such exists. It follows from the definition that if the series u diverges to + or - 00 , so F1180 will

k=1

H. The most interesting application of this method, then,

will be to series whose partial sums oscillate between

finite or infinite limits. For an example, let us consider the series

F1 + I - I + I - I + ... Then 81=a

a + 1

83 = s.

a + i

s =a 2 n-1

52n =a+1. Nov, H21_ 2n-1 2n-1

- 1 +T. + i u + n - i ¡ a-+1 ¡

2n - I (2n - i)Ça)Sa + i), ___________ 2ia-+ i

a +1J and lia H 2a(a +

2a+1

-1 (;

Also, H2 2n__ 2a(i. 1 ñ+ n

Thus, the given serLes is swnblo by the harmonic mean

to the value 2e(a + i).

From this result we can see that the haraonic

mean sum of a nonconvergent series is not necessarily

linear. That t, chancing a by an amount b does not

necessarily change the hmonic mean sum by b.

Let us investigate the result of interpolating an

infinite number of zeros in the series of the previous

example. Corwider the series

a+1+O-i+i+O-1+1+O- Then s=a+1

83n+1. =

= a + j Now 3n 3a'a + i)

n 2h

And H3n+i = ! (n i- 1)ja1a + i) n+1+ 2n )na+n+a+1

a

Thus lUn H31 3a(a + i) 3a+I Aleo 11 2 = -

3n+ 2 on + 2 (a)[ + 1 + n +I+ 2ii-:i:--I 3naÏ- n+ ¿a + 1

-i:--- ----:i:--i

- 3aIa + i) and 1imH32

The harmonic mean sum is now 3a(a + i). Hence, the a +-I--

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 in. which, after a finite (k)

number of terna, is recurrent with a fixed period (a)

of terms, and O < n K M, ni and M finite. Then

H n n

= .-. + ---+ . ..- +

For convenience, let (p1 + ... + and let

(pk1+...+pk+a)=A. ThenHk+ k+na fl8

= K+ii

and H k + na + d, where d is an integer such

that I d a, and D = p «. i- k +

D, then,

can take on at most a finite (a) number of values. 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

(c,i) under these conditions but not necessarily to the

same value, as we have seen in some of the previous

example s.

N0E: Suinmability by Cesro means does not neces-

sarily imply suimaability by iarmonic means, nor is the

reverse true. For example, consider the sequence

1' 2' 82n' 82n i. 1' ' where the pair 82n - i'

52n. is either 1, 1 or 1/10, 19/10. For n = 1, 2, ... (i) i , and

(i) = either I or 2m + 1/10

LU +

(i) Hence, liza C = I

Now, H _____________________ , n - I-:: I +-.; + I

'A n

+1 I 2 82n-1 82n,

2n

where the pair I , I is either 1, 1 or 10, 10

82n-1 82n

Then î. i.. th aritbmetic mean of n pairs which are 2n

either 1, 1 or 10, 10/19. The aritbnietic mean of 1, I

is 1, and that of IO, 10/19 is 100/19.

Construct si, 2' ' as follows: 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 con-

tinued, 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 possible to define higher orders of har-

monic means. A slight change in notation is necessary.

Let n . We have seen that i1' -

: ; i

1 8

-p , then -y 8. Then let

n n - + -F...+ 1?

Since 1H} is a sequence approaching the value S,

2i S. In general,

H_____n ______ r n -- rr. .

- r - r -

and rHn -, S

We may now state a definition of order. If r is

20

the smallest integer such that urn H exista, then n

u is summable (rH) by harmonic means of order r.

ki

From the above, it is apparent that if a series is sum-

mable (rH) it is sumniable (r +

1H) to the same value.

If u is convergent, it is sumrnable (rl!) for r k1

positive integer.

21

VI. TRE WEIGHTED HARMONIC MEAN

00 n

Cone ider the series . u, where s = u O. k=1 k=1

If the series is convergent, urn s S. Let I

Sn

Then P = urn p = urn I I . Consider n flPco

n

Cn , where

=a1+cn_1P2+ s..

n C = , and um O. Then

k=1 n-Ç +

en IP2 + +

w=____ --c-

and 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 = 1, this method reduces to

the ordinary harmonic mean.

2

VII. SUMMATION BY GEOMETRIC MEAN

fl

Again consider 2 u1 where s = u s - O.

k=1. n k=1 k

We may define a geometric conventional sum as

follows: Let G (8152...8)hl'fl. Then

liai (2152.. e)h/'1

= G, the geometric conventional sum.

Phis method is regular, for by section III we saw that

H G A, and if I Uk is ordinari1 convergent to S,

k=1.

H A = S. Thus, G also equals 5. However, G may

exist when S does not. In that case G is a conventional

sum for the series.

There are certain restrictions that must be

placed upon the geometric meen. Clearly, in order to

have a value different from zero, none of the partial

sums can be zero. Also, in order to avoid imaginary

results we must insist that the product of the partial

sums be positive for al]. n N, where N is a finite

integer.

For an illustration of this method, let us con-

aider our familiar series, a + i - i + i - i + ...

82n = a + I

52n + I a

I/2n 1/2

G2n = [a(a + 1)nj

[a(a + i)J

2

I n+1 ri

rn + 1 )fl]2fl+'l ri+L

+ . = La (a + I a a

1)2flI

1/2

Then, urn = ' 2n + i = [a(a + 1)7

fl-CK

Thui, the serles te suiuinable by the geometric mean

1/2 to the valuo lIa(a + l)J

VIII TRE WEIGHTED GEOMETRIC MKAN

00

Once again consider the series where k=í

n = and ali are greeter then zero. Consider

k=1

i

s1)7vhere the e1 and C ere

as defined in section VII. Then

in .

- + .

+ e1 ina

nd 11m in G = lin in s = in S, provided this exists.

Therefore, G S if the &eries is convergent to k=1

S, 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

to certain nonconvergent series. In a similar way we can

f md a method of evaluating infi1te products that is

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 TTa1 , and let p = a1a2...a.

Let G . Then we have

in G in p1 + in p2 + ... + in p

-- n

If e is convergent, thon the sequence (in )

converges to lnîíwherelTis the ordtnry value of the

infinite product. Hence, by arithmetic mean summation,

in G - in iT, or urn G =TF. However, if um p

does not exist, but um 0tl does exist, then urn 0ri may be n-ì°o

co

considered to be a conventional product of IT a,

Since this method is equivalent to that of Holder,

26

we may 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

(1)(2)(1/2)(2)(1/2)... . Now

p1 =1

p2 =2 ........... 2n =2

2n + ± = 1

Obviously, the product does not converge. flowever,

1/2 n

= (i)(2)(1)(2)...(2)] 21/'2, and

I n

02n + = [(I)(2)(1)(2)...(2)(1)] =

Then, uni = lu a2 + 21/'2. Therefore, 21'2 ja

the conventional product.

L FURTHER EXAMPLE$ OF TILE &RMOtIC MEhN METHOD

Consider the sertes

U=1_ I I

In tht8 SetieE, .pplyirg,

first ten terms to obtain

0.685. The aun of

I I I I I I I + r - + 7 - + +

tc hernicnic ìieri method to the

&n approximation, we find that

the first ten terms is 0.61456.

r

The velue 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

for which

U=2-5-2+7-5-2-i-7-5-2+7-

=2

83

... . 0

s. 2 fl

+ I = 2

83n + 2

Here the conditions of the result in section V concerning

series with oscilleting partial sums are satisfied with

a = 3 and A = (-1/5 + 1/2 - 1/3) - 1/30. Hence, we

expect the result lin = )/(-I/30) = -90. In order to )-3X L

28

check this', we find tht

3n -90, -n + n - n

+ I = - + -90, anc

+ 2 __n +2 , 11m

+ 2 = -90.

-n+1-1 n-'°o 33 Thus, the series is summb1e (1H) to the value -90.

XI. COMPARISON OF CESARO AND HARMONIC SUMMATION

Consider the series U = a, where s k=1 k=1

and D (s/ M, M finite. If the series is con-

vergerit, let its sum be denoted by S. Let

(i) C I + W + n , and let 4H n We may

n = -- - - I. - i 3_. 3

J- n

now state the following properties or characteristics of

Cesàro and harmonic summation:

I. If U 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).

3. If U diverges to + or - 00 then both the

arithmetic and harmonic means diverge to +aO or .00

k. IC U is suinniable (c,i) or (1H), it either converges

or oscillates between finite or infinite limits.

(i) (i) 5. If fl C1 = 11m Cr = ' and finite, thon

U is said to be bounded (c,i). Similarly, if n1n=i;, I11n' 'nd$ftnite, then n- fl-3

U is said to be bounded (1H).

6. The interpolation of a finite number of zeros does

30

not chníe either the Cesro or harmonic $um.

'T. The interpolation of an infinite number of zeros

my chance both the Ces&ro and harmonic sums.

8. 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

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