11 



SUM. SUMMATION'. 



. M-.MMAT10X. 



a beginner to tat result* the proqf of which must be reserved I where a,, a function of x, generates the several terms by making 



for a more advanced stage of his progress. 



1. When the terms of a aeries are alternately positive and negative, 

 aa in a - a. + a, Ac., the sum of the series ad i../m'rw may thus 

 be expressed [DirritHKM-r] (D. (.'., pp. 656-StH') : 



u 

 2 



A o 

 4 



AV. 



8 



AV 

 16 



which U frequently more convergent than the aeries itself ; in fact, the 

 less convergent the aeries is, the more convergent is the transformation. 

 l)r Mutton's method of obtaining the transformed series is as follows : 

 Take a number of the successive sum* a u , o,, o,, Ac., and let 

 = , 8, = o,, , B, = a,, o, , 8, - a a, + a. , Ac. 



Take the half sum of s, and s,, the half sum of s, and s, , the half sum 

 of s, and 8,, Ac. Let these be T , T, , T,, Ac. Kepeat the process : 

 take the mean of T and T,, that of T, and T,, IK., which call u,,, 

 ., Ac. Take the mean of v a and o,, that of u, and u,, Ac., which 

 call v u , v, , Ac. Then the set s,,, T , u,,, v,,, Ac., will severally approach 

 nearer and nearer to the series required ; in fact 



*= T. u o=- - 



A' : a 

 " 8 '** 



Aa 

 " 



It would however be somewhat easier to proceed as follows : having 

 formed differences as far as may be thought necessary, say up to 

 A'a,,, take half A"a from A"~',,, half the result from A'-^w 

 half the result from A""'a and so on until a has been used : after 

 which lialve the result ogam. In either cose we need not begin at the 

 beginning of the series : if it be more convenient to begin after a,,, let 

 A = a,, - a, +....- a, + o, , and calculate this separately : then 

 calculate a,, a,, + .... from the rule, and we have A 10 



5, + . . .) for the series required. The following is an instance from Dr. 

 utton (' Tract",' vol. i., p. 191 ), the series being 1 i + i i + 



Sum*. 



1 



0-5 



S33333 

 583333 

 783333 

 616666 

 759524 



634524 

 ~.rar,r. 

 745635 



45 



736544 

 653211 



The several orders of means. 



694878 



C92560 



693552 

 00001:0 

 692858 



693362 

 C92984 



9n 

 693205 



693131 

 aavina 

 693158 



693142 



The result is '693147, which is correct to the sixth place, and is 

 more than could be got from the series itself by actual summation 

 of a million of its terms. Dr. Hutton begins in forming the means 

 with 1 4 + ____ + j : we shall therefore try the other method, 

 beginning with !. 



Terms. 



142867 

 125000 

 111111 

 100000 

 090909 

 088883 



Orders of differences. 



11111 

 9091 



7576 



2ii2o 

 1515 



1190 

 758 

 506 



432 

 253 



17g 



- 89-5 

 432 



1-4 



, . . - i = -eier.ofifi 



0764795 



2)521-.'; 



260-8 

 -1190 



2)-1450-8 



-726-4 

 3968 



2)4!" 



2346-7 

 -17857 



2)-202087 



10101-9 



1 !_ 



2)1529589 

 76479-6 



Thia last process will lie found on trial the easier of the two. 

 J. The MUM of the series a,, o, + a a &c. </ may lie 



thus expressed (D. 0., p. 655) -. 



" 4 " 48 ~ 





 480 + 80640 ~ ic ' 



x = 0, 1, 2. Ac'., in succession, and ', a,,'" ' , &c., mean the values of 

 the odd differential co-efficients of a, when .r is = 0. Tins n. in- 

 formation is useful when the values just mentioned are n 

 sidurable. Another form, which is sometimes more convenient, i 



2l4] 



3..' 

 2[] 



l! . 

 2 [8] ~ Sc 



where [m] means 1.2.3 m. In the instance before n . :>n<l th >t 



legin from the same term as before, let 



1 , [3) [5] 



- o 72 ' 74 ' a " ~ ~ 76 > "" 



whence the series required from and after } is 



11 

 14 + 4.72 



1 

 2.4.7' 



8 

 2.C.7 



17 

 2.8.7* 1 



Call th.se term.* ^1), (2), &c., and begin with J + . 

 we have then 



6166666667 



(1) -0714285714 



6880952381 



(2) -0051020408 



6981978788 



(3)--0000520616 



(4) -0000021250 



69314734-J:! 

 (5)- -0000001843 



6931471580 

 True Answer -6931471806 



The result of this comparatively easy process is us correct as the 

 summation of fifty millions of terms of the series. 



3. The sum of any large number of terms of a scries may be f< mini 

 by summing the whole series ad in fin it am, then doing the same with 

 the terms following the last term which is to be retained, and sub- 

 tracting the second result from the first. 



4. The sums of such series as are included under 1""" + 2~" + 3~ + 

 &c., such as 



11 11 



continued i"l in uiiiii'nt, may be given for reference in the fi.ll.iwin;; 

 table. The first term will presently be explained. JW,- will lie 

 found in D. C., p. 554. 



1 



2 



3 



4 



5 



6 



7 



8 



9 



10 



11 



12 



Sum of Scries. 

 -5772156649016329 

 1-6449340668482264 

 1-2020569031595943 

 1-0823232337111382 

 1-0369277551433700 

 1-0173430619844491 

 1-0083492773819-227 

 1-0040773561979443 

 1-0020083928260822 

 1-0009945751278180 

 1-0004941886041194 

 1-0002460865533080 



log oo 



The first line means that the sum of the series is infinite, but that 

 t'ie e\|n-eiwion for a large number of terms contains the logarithm of 

 that number, which being removed, the rest of the exjn 

 approximates as the number of terms increases, to -677-15 .... 



5. The series I* - Z~* + ... is connected with 1" + 2" + . . . by 

 the following simple law : 



6. The sum a + a, + a a + . . . '/ tnjhiltum may be thus 

 formed : 



/ 

 / 



1 1 ' 1 '" 1 ' 



<,<** f 2 o- c T + 30 W - 42 [Sj + 





where the notation U as previously explained, and g, j^, &c , are the 

 series of NUMBERS OP BERNOULLI. To apply this, for example, to 



