DIVERGENTAND CONDITIONALLY CONVERGENT SERIES WHOSEPRODUCT IS ABSOLUTELYCONVERGENT*
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
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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.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 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~^A}' 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=715=71 — 1 /
p-0 \ 4n —4p 4p -f 1 4n —4p —3 4p -f 4 1 1 + 4w— 4p Ap + 1 4w— 4p —3 4p> +■ 4J 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=0v* -4^4-14^-fl 4n —4p 4p + 4 1 4n — \p — 3 4p + 1 4n — 4p 4pïTï)—' where n = %n — 1 or %(n — 1) > according as n is even or odd. When n is odd,
_____ 1 1 1 1 1 1 1 a _ 2?i + 2 2«,+ 2 ~~2w-l 2w- 1 ~ 2n + 2 2n+2~ ~ 2n-l 2n-l;
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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 -<7——-,-11 4p + l (4w— 4p -f-l)(4/i —4jo — 3) 4/>-f 1 n
It is easy to see that 2k 2 2C-< a log (2n), n=i n
where a is a constant greater than 3/2 log 2 . Hence
547TÏ 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. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1901] SERIES WHOSE PRODUCT IS ABSOLUTELY CONVERGENT 29 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_ ^o\ 4n —4p + 4 4p + 1~ 4n —4p + l 4p + 4 4w —4p 4p + 1 4w —4p — 3 4/>Ti)"*' 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 1 2/1 + 1 2/i + 1^2/ii o„ + i 4 I 2wO™ + i 4,4 ^ l 2/1 + 1 2/i + 1 ' License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 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_M 1_ ¿_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 V6>"+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-6p6> + 3 6>46^ + 1 ^6w-+ fi„ 6>ß„ + i 2O 6j>ß™ + i 2o T6w-6>+ + 1 6^ + 3 1 _j._ _1_ 1 1 1 \ + QnT—QJ)+~3 "6>TÏ ~"6n - 6j9+ 2 6>+l! ~ 6n - 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 6n —6p —5 6/>+ 1 + 6»i5-ñ-0— 6p —3 5—r-H6jj + 2 + 6?i—6p —4 6jo+ 3 p=n' / 11 11 11 M + a--_Qn- 6p T7+ 1 5-rrQp + 1 + 6«_-5-6 - 6p - 3 6>-f«—7^ 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 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1901] SERIES WHOSE PRODUCT IS ABSOLUTELY CONVERGENT 31 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). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 32 F. CAJORI: DIVERGENT AND CONDITIONALLY CONVERGENT [January 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 : y^u "r( 1 _-_ *_i_ _\ ^"- ieíW.+ ai 4/j4-l4-a2 àp+2 + as 4p + 3+aJ and 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. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1901] SERIES WHOSE PRODUCT IS ABSOLUTELY CONVERGENT 33 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 , (II) + {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 + #} — iV„ ■i+^}-K+^'}- , + l + ß"'\-{vX2 + 2 + ß"3 0. 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 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 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} + {«„, - 2 + ft} - {v + ft"} - {vX2p+ 2 4. /.;"} = o, 00 •_*- 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 Í {«12, - 7 + «,} + {u1%>- 4 4- „;} _ {Wi2p_ 1 + a'4) - [uX2p+ 2 + a"3} = 0, (VI) {«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 : {«12,-9+ #i} = {«„,-3 +ft}, K--6 +ft} = {_„_+ #'}, (VU) H2, 8 + rft} = Ky-2 + ft}, {vl2p-b + ß'2} = {vX2p+ l + ß"2}, {«i2,- 7 + ft} = {vx2p- 1 + ft}, {vX2p- 4 + ft} = {„ + 2 + ft"} . The same considerations applied to equations VI yield the following six rela- tions : {Mi2, - 8 + a,} = {uX2p- 2 + a'A , "12, 5 + ttJ = {«12,12, + 1 + <} 1 1 + a2} = {uX2p-l+a'i}, {«__ + a¡ {«,2, + 2 + a"3 (VIII) "12, 12, Mi2,-6 + a3} = {«_, + <'}> [U 12, ' 3 + «2}={«12, + 3 + a'; Applying relations VIII, one may write the series _^w_ thus, the parentheses being suppressed : __>«= E({«i2, - 8 + «,} + {uX2p-l + a2}- {uX2p- 6 -f «J ,=i - {«i2, - 5 + o4} + {«,., - 4 + a[} + {M12p_ 3 + a'2] + {Mi2,-5 + a<\- "12, ■4+<}-K*-8+«;}) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1901] SERIES WHOSE PRODUCT IS ABSOLUTELY CONVERGENT 35 Applying relations VII, one may write the series £«m thus, the parentheses be- ing suppressed : 2>„^i:(Np-9+0.}p=i + Kp 8 + /32}-{%,-7 + £3} + {« -6+/3;} + {«i2i-5 + /32}-{«i2î,-4 + /33} (X) + {«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' ■'Up 9 + ^l}-{«12„-12P + 9 + S {«12,-8 + ^2} E(+K»-i%,+i61 + {«.2„-i2J,+8 + + «;} «, «.2p-7+^} +K„-12P+7+ «4] «.2,-6 + /3¡} ■6+ Ä1 + K + {«12n-12, + 6 + «3} 12, 12n-12, + 5 + a2 «i2,-4 + /3¡} [u 12n-12, + 4 + aJ 12, 9 + ßx} + {uX2n_x2p + 9 + a'2} -8 + /32} 12n-12r-K8 + ai ^12, ' -7 + ^3}-{Wl2„_12;) + 7 + a4} — f«„l 12n—12, ,•> + ' 6 +' a 'l2, -5 + ^}-{«i2„_i2? + 5+a2} -4 + /33}) +j>=i E(+{«: P + 15 + /3, \u i2P-8 + «i}-{«12n-i2P+8 + ^ i2,-7 + «ä} - {«12„-12, + 7 + ß'i ■6+ «.,}-{« 12, 1271-12, 12p -5+aJ - {«12»-12, + 5+^3 -3 + «;} "12, -4 + a;} + K2„_I2p+4■p + ^2] 12, — V "12, -?+«2} P + Z + ßi} 8 + aJ + {v12/1-12, + 8 + /33} 12, 'I2p + {«12„-12, + 7 + ß2 _ 6 + a3] + {t,12n-12, + 6 + /3;} 12, 5 + aJ -3 + <}) + {« 12»-12, £3} ,-4 + a'x}-{v 12n-12, + 4 + ft} I2p = -12£{« 9 12}> A} 12n-12, + 16 + «,) 12n—12, + 4 + «1] ,=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 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 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. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use