DIVERGENT AND CONDITIONALLY CONVERGENT ... AND CONDITIONALLY CONVERGENT SERIES WHOSE PRODUCT IS ABSOLUTELY CONVERGENT* BY FLORIAN CAJOKI 1. Introduction. It has been shown by Abel

DIVERGENT AND CONDITIONALLY CONVERGENT SERIES

WHOSE PRODUCT IS ABSOLUTELY CONVERGENT*

BY

FLORIAN CAJOKI

§ 1. Introduction.

It has been shown by Abel that, if the product :

71—«

I(¥» + V«-i + ■•• + M„«o)>71=0

of two conditionally convergent series :

71—0 71 — 0

is convergent, it converges to the product of their sums. Tests of the conver-

gence of the product of conditionally convergent series have been worked out

by A. Pringsheim,| A. Voss,J and myself.§ There exist certain conditionally-

convergent series which yield a convergent result when they are raised to a cer-

tain positive integral power, but which yield a divergent result when they are

raised to a higher power. Thus,

71 = «3

Z(-l)"+13,7>=i n

where r = 7/9, is a conditionally convergent series whose fourth power is con-

vergent, but whose fifth power is divergent. || These instances of conditionally

* Presented to the Society April 28, 1900. Received for publication April 28, 1900.

fMathematische Annalen, vol. 21 (1883), p. 327 ; vol. 2(5 (1886), p. 157.% Mathematische Annalen, vol. 24 (1884), p. 42.§ American Journal of Mathematics, vol. 15 (1893), p. 339 ; vol. 18 (1896), p. 195 ;

Bulletin of the American Mathematical Society, (2) vol. 1 (1895), p. 180.|| This may be a convenient place to point out a slight and obvious extension of the results

which I have published in the American Journal of Mathematics, vol. 18, p. 201. It

was proved there that the conditionally convergent series :

V(_l)M-lI (0<r5|>).Sí n

when raised by Cauchy's multiplication rule to a positive integral power g , is convergent whenever

(î — l)/ï <C r ! out the power of the series is divergent, if (q— 1 )¡q > r. The case for which

(q — \)¡q = r was left untouched, but an examination of the formulae appearing in that article

readily yields the result that in this case the series is divergent. Hence the condition for diver-

gency should be written (q — l)jq = r .

25

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26 F. CAJORI: DIVERGENT AND CONDITIONALLY CONVERGENT [January

convergent series, yielding powers of the series which converge less rapidly the

higher the power and which for a sufficiently high power yield a divergent re-

sult, suggest the following questions : Does the product of two conditionally

convergent series never converge more rapidly than do one or both of the factor-

series ? Can the product of two conditionally convergent series or of a condi-

tionally convergent and a divergent series in no case be absolutely convergent ?

The first doubt of the correctness of a negative reply arose in connection

with the conditionally convergent series :

»=» (_ iy+i

¿=i n

This yields convergent results, no matter to how high a positive integral power it

may be raised.* If the series is squared, the constituents, l/(n — r) (r -\- 1),

which go to make up the nth term of the product, namely,

r=n-l 1 1

£o n — r r + 1 '

are all positive or all negative for that term. Hence the numerical value

of the nth term is the sum of the numerical values of all the products

l/(ra — r)(r + 1) . If the square of the series is indicated by

»1=1

then all the constituents, an_T • l/(r + 1) , which enter into the composition of

the 7¿th term in the cube of the series have all like signs in that term. Simi-

larly for higher powers. Now it has been shown that all the positive integral

powers of the conditionally convergent series :

n=°o (_ 1^"+I

Si n

continue to converge, even though each term of a given power of the series is

numerically the sum of the numerical values of its constituents, and it is evi-

dent that, if the constituents of each term were partly positive and partly neg-

ative, each term would be numerically smaller. Can the signs of the terms of

two conditionally convergent series and the numerical values of its terms be so

chosen that the product of the two series is absolutely convergent ?

In an important article on the multiplication of conditionally convergent

series-j- A. Pringsheim states that he knows no a priori reason why there might

not be special cases in which such a product is absolutely convergent. " Aller-

*American Journal of Mathematics, vol. 18, p. 204.

f Mathematische Annalen, vol. 21, p. 332.


1901] SERIES WHOSE PRODUCT IS ABSOLUTELY CONVERGENT 27

dings," he continues, " war es mir bisher nicht möglich an irgend einem bestimm-

ten derartigen Falle die unbedingte Convergenz zu constatiren, und es dürfte dies

— wenn es überhaupt solche Fälle giebt, was ja immerhin als fraglich erscheinen

muss — wegen der zusammengesetzten Beschaffenheit der Reihenglieder wv eben

nur sehr schwer gelingen."

§ 2. Two typical examples.

We have succeeded in devising special cases in which an absolutely convergent

series is obtained as the result of multiplying two conditionally convergent series

together, or one conditionally convergent series by a divergent series. The

series obtained by removing the parentheses from the series :

f=n( 1 1 1 1 \

Sl~ ]^\Tp~^~±p^^ +ïp~^\~ïp~Â}'

p=" ( l l 1 l \

^2= 5(v47TT + 4i? + 4~ 4^ + 1 "4^ + 4,)'

are both conditionally convergent. Let the four terms obtained in either series,

or the terms in a similar series, for any one positive integral value of p, be

called a " set " of terms. The 4w-th term of the product of Sx and S2 is

p=71 — 15=71 — 1 /

p-0 \ 4n — 4p 4p -f 1 4n — 4p — 3 4p -f 4

1 1+ > + 4J4w — 4p Ap + 1 4w — 4p — 3 4p ■

We see that the sum of each set of constituents for the 4«-th term is identi-

cally zero ; therefore the entire 4n-th term is identically zero.

The (4n -f- l)-th term of the product series is as follows :

111

¡h\én~^áj4p + 1 é.p + 1 4n — 4p 4p + 4

1+

4n — 4p — 3 4p + 1 4m — 4p 4p -M)

*ë?7 1111

p=0 v* -4^4-14^-fl 4n — 4p 4p + 4

1

ïTï)—'4n — \p — 3 4p + 1 4n — 4p 4p

where n = %n — 1 or %(n — 1) > according as n is even or odd. When n is odd,

_____ 1 1 1 1 1 1 1a _ 2?i + 2 2«,+ 2 ~~ 2w-l 2w- 1 ~ 2n + 2 2n+2~ ~ 2n-l 2n-l;



and when n is even,

2/i + l 2n + l'

In the first summation we are adding 4n' -f 4 constituents from one end of the

column or series yielding the (4n + l)-th term of the product ; in the second

summation we are adding 4/j' -f 4 constituents from the other end. In the

middle of the series the sets of constituents will overlap, if n is odd, and will

not touch if n is even. When n is odd, then a represents the sum of the consti-

tuents which have been counted twice ; when n is even, then a stands for the

negative value of the constituent not counted at all. Each of these constituents

is of the degree — 2 with respect to n and as their number is finite, their sum

cannot be of a higher degree with respect to n than the degree — 2.

The (4«,-)- l)-th term, simplified, is

p=n'/ 2 _ 4 \

2Sv47+T (4«-4^ + l)(47i-4p-3)J-a-

For any positive integral value of n the fraction 4/(4re — 4p + l)(4?i — 4p — 3)

has its maximum value when p = n'. Writing n fovp, we obtain a fraction

which is less than 1 /n1. Hence for any value of p,

1 4 11-<7——-,-

4p + l (4w — 4p -f- l)(4/i — 4jo — 3) 4/> -f 1 n

It is easy to see that2k 2

2C-< a log (2n),n=i n

where a is a constant greater than 3/2 log 2 . Hence

and

547TÏ<alog(2w)'

PSi! 1 4 e1 1 1 alog(2re)

¿4^1 (4n-4p + l)(4n-4p-8) < ¿47+1 ñ2 < n1 '

But a number m can be found, such that, when ?i~> m , we have

a log'(2/i) 1

w2 /i (log w)s '

where Ö > 1 . Therefore, for n > m , the (4/i + l)-th term is numerically less

than the nth term of a series known to be absolutely convergent.



The (in + 2)-th term of the product-series is

«?/ i_ i i_i

¿¿ \~ An — 4p + 4 4p + 1 4n — 4p + l 4p + 4

1 _1_ 1_1 \

4/i — 4p 4j» -f 1 ~~ 4w — 4/> — 3 4p-\-4J

'-7 1_1_ 1_1_

ô\ 4n — 4p + 4 4p + 1~ 4n — 4p + l 4p + 4

Ti)"*'4w — 4p 4p + 1 4w — 4p — 3 4/>

where »' is defined as before and where

11 11= 0,

2/1+2 2/1-1 2/1-12/1+2

when w is odd, and

1 1 _1_ _1_a_ +2/1 + 4-2/1 + 1 2/1+1 2/1 + 4,-'

when n is even. We see that the entire (4n + 2)-th term is identically equal to

zero.

The (4n + 3)-th term of the product-series is

«7 1_1_ 1_1¿Í\+ 4/i — 4p + Í 4/T+Ï — 4n — 4p + 4 4p + 4

1_1_ 1 1 \

~ 4n — 4p + 1 4jp +1 4/1 — 4/) 4p + 4)

^'/ 1_1 1_1_

+¿ \ 4n — 4p + l 4^ + 1 — 4n — 4p + 4 4p + 4

111

4n — 4p-\-\ 4p + 1 4« — 4/> 4p + 4

= à V4F+4 (4/1-4i9+4)(4»-4;/)J ~a'

vhere

1 1

2/1 + 2 2/i + 2'

when n is odd, and

1i o„ i I O™ i ,4 l2/1 + 1 2/i + 1^2/i + 4 2w + 4 ^ 2/1 + 1 2/i + 1 '


30 F. CAJORI : DIVERGENT AND CONDITIONALLY CONVERGENT [January

when n is even. The reasoning which we applied to the (an + l)-th term will

show that this term, too, when «>m, is numerically less than l/n(logn)e,

0>1.We have now shown that the 4?i-th term and the three immediately succeed-

ing terms are either identically zero, or, for «. > m, numerically less than

l/n(\ogn)e. Hence the product of the series Sx and S2 converges absolutely.

Again, take ■ the conditionally convergent series, obtained by dropping the

parentheses in the series :

T .W_i_- + x_L_ + _i_M1_ ¿_í \6_> + 1 6_> + 2 6_> + 3 6p + 1 6p + 2 6/j + 3/

and the divergent series, obtained by dropping the parentheses in the series :

vV 1 1 ^L_ _1_ ^L_ _1_\2= ¿Í V 6>"+l + 6> + _ ~ 6p~+3 + 6/> +1 + 6p~+2 ~ 6p~+Sj-

The 6«-th term of their product, viz.,

p=n—\7 = 7.-1/ J ^

5 V 6n-6p + 3 6>4 + fi„ ß„ i O ß™ i o +6> + 3 6^ + 1 ^6w- 6> + 2 6j> + 2 T6w-6> + 1 6^ + 3

1 _j._ _1_ 1 1 1 \

+ QnT—QJ) +~3 "6>TÏ ~" 6n - 6j9 + 2 6>+l! ~ 6n - <op + 1 67+~3/'

is seen to be identically zero.

The (6ra + l)-th term of the product of Tx and T2 is

f.u_-_-_-¿SV^6n-6p-|-l 6/j + l 6»-6 6^ - 3 ñp + 2 6w - 6p - 4 6^> + 3

1+ 5-ñ-0 5—r-H +

p=n' /

M

6n — 6p — 5 6/> + 1 6»i — 6p — 3 6jj + 2 6?i —6p — 4 6jo + 3

11 11 11+ a--_ T7 5-rr + _-5-6 «—7^ +Qn - 6p + 1 Qp + 1 6« - 6p - 3 6>-f 2 ' 6_ - 6> - 4 6> + 3

_1_ _1__1_1_1_ 1 \6n-6p-6 6/> + l 6ra-6>-3 6> + 2 6rc-6>-4 6j_ + 3/ _a

_ *=%/ 1 -6 \~ èi\6>+"ï (6n-6> + 1)(6»-6^35)J _a'

where w' is defined as before and where



1_1_ 1 1_ 1 1 1 1_ 1 1_a~ 8w 8» — 1 3/^Tl 3» 8» —2 3ra^2 + Wn3» -1 + 3/i-l 3w

1 1~~ 3/1-2 3w —2'

when w is odd, and a = — 1/(3« + l)(3/t + 1), when n is even. By repeating

the reasoning used in previous cases we find that this term, for n > m, is

numerically less than l/zi(log nf, 6 > 1.

The (6»+3)-th, (6» + 4)-th, and (On + 5)-th terms in the product of Tx

and .Z2, when subjected to this course of reasoning, all yield the same conclu-

sion. The (On + 2)-th term is identically zero. Hence the product-series is

absolutely convergent.

§ 3. The general method.

The method by which the above factor-series were constructed is simple. The

harmonic series was modified by adopting some simple law of succession of signs.

To render the series conditionally convergent, make the number of + terms in

a set equal to the number of — terms. If these numbers are unequal, then the

series is divergent. Then add suitable finite numbers to the denominators of

some of the terms in the set, so that, when the product of the two series is

formed, the constituents of any set of the nth term in the product, when added,

yield a fractional sum having denominators whose degree in n is higher by two

than the degree of the numerators. We were able to prove the absolute con-

vergence from the fact that in each complete set of constitutents of the nth term

in the product-series there are as many positive constituents ras there are nega-

tive ones. To secure this even distribution of signs among the constituents for

the sets in every term of the product, it is necessary that in one of the factor-

series the number of + terms in any set be equal to the number of — terms in

that set. This condition is not sufficient, because among the terms in the prod-

uct some might have a preponderance of + constituents and others a prepon-

derance of — constituents. This happens, for instance, when we multiply the

series +-|-1--|-etc. by itself.

In the proper selection of signs the following rule, which we give without

proof, may be of service. Let a, denoting the number of terms in each set- of

the series £wm, be an even number and equal to 2c; also let the first c terms

in each set be + , the remaining terms — . Let the number of terms in each

set of the second series, ^,vm, be o. Then, in order to secure as many + con-

stituents in each set of a term of the product of the two series S«m ar*d Sv ,

as there are negative constituents, it is sufficient that d should be a factor of c,

where d is the smallest value, numerically, satisfying the congruence 6 = d (mod a).



In explaining the further process of constructing our series we shall start out

with a particular selection of signs, but the process is the same as for any other

selection, similarly made. Let a = 4, 6 = 3, then c = 2 and d = — 1, a

factor of c . Accordingly, the signs of terms in each set of the first series are

+ H-. Let the signs of the b terms in each set of YLv be + -)-.

We have now the two factor-series obtained by dropping the parentheses from

the series on the right hand sides of the formulas :

and

yû "r( 1 _-_ *_i_ _\^"- ieíW.+ ai 4/j4-l4-a2 àp+2 + as 4p + 3+aJ

r="( 1 1 1 \2>»'= g W+Ä + W+i + ß~2 ~ äp + 2 + ßJ '

in which the Greek letters are quantities to be chosen, if possible, so that the

product of the two series shall be absolutely convergent.

For brevity, we shall, in the series SMmi represent by {ua + a] a fraction

l/(ap + a), and, similarly, in the series _j«m i Dy {vb -\- ß} a fraction 1 ¡(bp + ß).

Accordingly we have, dropping the parentheses as before,

Z».SI((% + «i} + {% + 1 + **} - KP + 2 + a3} - {« + 3 + a4})p=í

and

E«m-E({«3, + r51} + K + i + /3j-{^ + 2 + ^}).p=l

In the product of __wm an(i __«m7 a set w^l consist of 12 constituents. The

first set in the (12w)-th term will be (since {vX2— 9 + ßx} = {v3+ /3,} , and

similarly in other cases)

- {"i2„ + 3 + <} K - 9 + /3J - {«^ + 2 + „;'} {.u - 8 + ß2}

- {ul2n + 1 + a"2} {vX2 - 7 + /3J + {uX2n + <} {_u- 6 + ß[}

(I)+ í«u.-8 + a;} {«„_ 3 + #} + {Mi2n_4 + «¡} {%- 2 + /3'2'}

+ {«_h-6 + aJ {«.2-1 + #} - {ul2n- 6 + a3} {v12 + /.;"}

+ {•«, - 7 + a2} {.u + 1 + /3'2"} _ {«^ - 8 + a,} {vl2 + 2 + ^"}

Here each of the quantities ßx, ß2, ß3 occurs four times. We have broad-

ened our assumptions by attaching a single accent to each letter the first time

it is repeated and a double and triple accent the second and third time it is re-

peated. Accordingly ßx, ß[, ß'[, ß"x , ß2, ß'2, ß'%, ß"2 , ß3, ß'3, ß'3, ß'3" may

in general represent each a different number. Similarly, the letters a{, a2, a3, a4

receive a single accent the first time they are repeated in a set of constituents

and a double accent the second time they are repeated.



We shall limit ourselves to the case where, in any term of the product series,

each set, when multiplied by n, yields a product that converges toward zero

when n = co. Accordingly, if the set I be multiplied by n, we get, for

n = œ (supposing the Greek letters to be positive constants, or variables which

approach positive finite limits), the first of the two following equations :

-K-9 + A}-K-8 + /3J-{«u-7 + /9J + K-6 + /S;}- K-5 + ß2} + {vX2-4 + ß'3} + {„„-8 + ß'i} + K-2 + ß"2}

+ K - 1 + ß'S) - {«12 + ß"ll + {«12 + 1 + ß'il - {«12 + 2 + ßl'} « 0 ,

+ {Vu-9 + ßl}-{vX2-8 + ß2} + {vX2-7 + ß3} + {vX2-0 + ß'x}

+ {«12-5 + ß'2] + K-4 + /33} - K-8 + /3';} + K-2 + #}

(II)

— iV„ , + l + ß"'\-{vX2 + 2 + ß"3 0.■i+^}-K+^'}-

The second equation in II is got from the (12« + l)-th term by the same

procedure. All the even terms between the (12/i)-th and the (12«. + 12)-th

yield the same equation as the (12/i)-th term ; all the odd terms yield the same

equation as the (\2n + l)-th term.

The last set in the (12«)-th term of the product is

- {«i2„ + 2 + /3'3"} {uX2 - 8 + «J + {vX2n + 1 + ß'2} {uX2 - 7 + «,}

- {«i2„ - 1 + ß"i) {«i2 - 6 + «3} + {«.2,- 1 + ßl) {«12 - 5 + «J

+ {«i2„- 2 + /3'2'} {«,2-4 + a'x] + {vX2n- 3 + /3';} {«u_ 3 + <}(III)

■4 + /3;}{«12-2 + a; -5 + /3;}{«12-l + <}

+ {«12n- 6 + ß[} {u12 - a"x} _ {vX2n - 7 + /33} {«12 + 1 + a'2}

I - {vX2n- 8 + /32} {«12 + 2 + •;} - {vX2n - 9 + /3,} {Ml2 + 3 + <}.

Assuming set III, when multiplied by n, to yield a product which converges

toward zero, when zi = 00, we get the first of the following three equations :

- {«12-8+ai} + {«12-7+a2} - {«12-6 + a3} + {«12-5 + aj

+ {ul2-4 + a[} + {Ml2_8 + a;} + {Ml2_2+a3} - {«u — 1 + <}

+ {uX2 + a"x) - {uX2 + 1 + a"2} - {m12 + 2 + a"3) - {uX2 + 3 + a'4'} = 0,

+ {M12-8+a,} - {ul2-7 + a2} - {Ml2-6 + a3} - {uX2 - 5 + aj

(IV) - {«12-4 + 0 + 3 + a2}-{Ml2-2 + a3} + {Ml2-l + a¡}

+ {uX2 + a"x\ + {uX2 + 1 + a"2) + {m12 + 2 + <} - {ux2 + 3 + <} = 0 ,

+ [uX2 - 8 + ax] + {w12 - 7 + a2} + {m12 - 6 + a3} - {«12 - 5 + «J

+ {/i12 - 4 + a;} - {m12 - 3 + a;} - {uX2 - 2 + a;} - {uX2 - 1 + «¡}

- {«,2 + <} + {«,2 + 1 + "'2'} - {«,2 + 2 + o'j'} + {uX2 + 3 + a'4'} = 0 .

The second and third equations in IV are derived from the last set in the

(12«+l)-th and (12n + 2)-th term, respectively. The (12zi + 3)-th,

Trans. Am. Math. Soc. 3


34 F. CAJORI : DIVERGENT AND CONDITIONALLY CONVERGENT [January

(12/i + 6)-th, and (12n+9)-th terms yield the same equation in IV as the

(12_)-th term; the (I2n + 4)-th, (12n + 7)-th, (12re + 10)-th terms the same

as the (12«4-l)-th term; the (12« + 5)-th, (12_ + 8)-th, (12w + H)-th the

same as the (12n + 2)-th term.

The equations II, when simplified and by the substitution of vi2 for vXi

made to apply to other sets in the terms of the product, are

i-fo»-8 + £,} + {iV-6 + 0;} + K_ -4 + /33}

00+ {«„, - 2 + ft} - {v + ft"} - {vX2p + 2 4. /.;"} = o,

•_* - 9 + ßx) + {«,2, - 7 + ft} 4- {%, - 5 + ft}

- {«i2, - 3 + ft} - {v12p - 1 + ft'} - {vl2p + 1 + ft"} = 0 .

The equations IV, when simplified, uX2 being written for uX2, become

(VI)

Í {«12, - 7 + «,} + {u1%> - 4 4- „;} _ {Wi2p _ 1 + a'4) - [uX2p + 2 + a"3} = 0,

{«12, - 8 + aj - {uX2p - 5 + a A - {uX2p - 2 + a'3] + {ul2p + 1 + a"2} = 0 ,

{«12, - 6 + aa} - {«12, - 8 + «_} - {«„, + <} + {w12p + 3 + a"4) = 0.

If we aim to assign to the Greek letters constant values, then, in every equa-

tion in V each fraction must admit of being associated singly with another fraction

in that equation which is numerically equal to it, but of opposite sign. This

relation is brought about by the following six equations :

(VU)

{«12,-9+ #i} = {«„,-3 +ft}, K--6 +ft} = {_„_ + #'},

8 + rft} = Ky-2 + ft}, {vl2p-b + ß'2} = {vX2p + l + ß"2},

{«i2, - 7 + ft} = {vx2p - 1 + ft}, {vX2p - 4 + ft} = {„ + 2 + ft"} .

H2,

The same considerations applied to equations VI yield the following six rela-

tions :

(VIII)

{Mi2, - 8 + a,} = {uX2p - 2 + a'A , "12, 5 + ttJ = {«12, + 1 + <} 1

"12, 1 + a2} = {uX2p-l+a'i}, {«__ + a¡

Mi2,-6 + a3} = {«_, + <'}> [U

12,

12, '

12,

{«,2, + 2 + a"3

3 + «2}={«12, + 3 + a';

Applying relations VIII, one may write the series _^w_ thus, the parentheses

being suppressed :

<IX)

__>«= E({«i2, - 8 + «,} + {uX2p -l + a2}- {uX2p - 6 -f «J,=i

- {«i2, - 5 + o4} + {«,., - 4 + a[} + {M12p _ 3 + a'2]

- {Mi2, - 8 + a,} - {m12j) - 7 + a2} + {w12p - 6 + aj

+ {Mi2,-5 + a<\- "12, ■4+<}-K*-8+«;})



Applying relations VII, one may write the series £«m thus, the parentheses be-

ing suppressed :

(X)

2>„î:(Np-9+0.} + Kpp=i+ {«

8 + /32}-{%,-7 + £3}

-6+/3;} + {«i2i-5 + /32}-{«i2î,-4 + /33}

+ {«12,-9 + ßi} + {«„-S + /32}- {vX2p-l + ß3}

+ {vX2p-6 + /3;} + {v-5 + ß'2) - Kp-4 + #}).

Series IX is conditionally convergent ; series X divergent. In both series

any constant positive values may be assigned to the Greek letters. The product

of the two series is absolutely convergent. This may be shown by examining the

12/i-th term and the eleven immediately succeeding terms of the product-series

in the same way as we did the terms in the product of Sx and S2. We shall

here write down the expression for the (12« + l)-th term only, viz.,

p=n'

E(+K»-i%,+i6 + «,1 + {«.2„-i2J,+8 + «;}

+ {«12n-12, + 6 + «3}

[u + 4 + aJ12n-12,

12n-12r-K8 + ai

P + 15 + /3,

— f«„ ,•> + 6 + al 12n—12, ' '

+ E(+{«:j>=i- {«12„-12, + 7 + ß'i

- {«12»-12, + 5+^3

— V

+ {«12„-12, + 7 + ß2

+ {«

P + Z + ßi}

£3}

9

12»-12,

■'Up 9 + ^l}-{«12„-12P + 9 + S

«.2p-7+^} +K„-12P+7+ «4]

12,■6 + Ä1 + K + 5 + a2

12,

^12, '

'l2,

\u

12n-12,

9 + ßx} + {uX2n_x2p + 9 + a'2}

-7 + ^3}-{Wl2„_12;) + 7 + a4}

-5 + ^}-{«i2„_i2? + 5+a2}

i2P-8 + «i}-{«12n-i2P+8 + ^

12,■6 + «.,}-{«

1271-12,

"12, -4 + a;} + K2„_I2p+4 + ^2]

"12,

'I2p

8 + aJ + {v

_ 6 + a3] + {t,

,-4 + a'x}-{v

■p

12/1-12,

12n-12,

12n-12,

+ 8 + /33}

+ 6 + /3;}

+ 4 + ft}

{«12,-8 + ^2}

«.2,-6 + /3¡}

«i2,-4 + /3¡}

-8 + /32}

-4 + /33})

i2,-7 + «ä}

12p

12,

12,

12,

I2p

-5+aJ

-3 + «;}

-?+«2}

5 + aJ

-3 + <})

= -12£{«,=1

12}> A} 12n-12, + 16 + «,) 12n—12, + 4 + «1]

- 12Z {«12,- 8 + ax} {vX2n_X2p + 15 + /3J K„ + 8 + /3J -a,,=1

where n = \n when n is even, and «' = 1(« + 1) when n is odd, and where

a is used in the same way as before. In this case a consists of one constituent

when n is even, and of eleven constituents when « is odd. By the very same

reasoning as that applied in the series Sx and S2 it follows that the (12« + l)-th


36 F. CAJORI : ON THE MULTIPLICATION OF SERIES

term of the product of IX and X, for n > m, has a numerical value less than

l/«.(log n)e, where 0 > 1. The two factor-series IX and X differ from the pairs

8X, S2 and Tx, T2 in this—that (suppressing in each series each parenthesis

which encloses a set) corresponding terms in IX and X need not have the same

numerical values. However, the three pairs of factor-series have in common

one property, to which allusion has not yet been made ; viz., in one of the series

of each pair, the sum of the terms in each set is identically zero, so that the

sum of the series is itself zero.

Colorado College, Colorado Springs.


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DIVERGENT AND CONDITIONALLY CONVERGENT ... AND CONDITIONALLY CONVERGENT SERIES WHOSE PRODUCT IS ABSOLUTELY CONVERGENT* BY FLORIAN CAJOKI 1. Introduction. It has been shown by Abel