VOL. LXXIV.] PHILOSOPHICAL TRANSACTIONS. 5Q5 



+ + &c), 2. + /, where / is a whole number; but by this method are introduced 

 into the denominator the factors 2z + /, 3z + /, &c. and nz -j- /, or which may 

 be reduced to the same (z-| — ) X n. If n successive general terms of the serieses 

 arising from Mr. Bernoulli's addition or subtraction be added together, and in the 

 quantity thence arising for z, the distance from the first term of the series, be 

 substituted nz, there will be produced serieses of the above-mentioned formula. 



1 1 . Multiply two converging serieses a + bx + ex 1 + dx 3 + &c. = s and 

 a + fix + y x ' 2 + & c - = v > or mi d any rational and integral function of them, 

 and the series resulting will be finite and = s X v, &c. Let a. + fix + yx 2 -j- 

 &c. x m = v be finite, and the resulting series will be finite and = s X v, &c. If 

 s be a series converging or not, whose ultimate terms are less than any finite 

 quantity, then will the series (a Ar bx + ex* + &c.) X (* + fix + yx 3 -(- &c. x™) 

 = v x s be a converging one, if * + fix -\- yx 2 + . . . &c. x n = O ; which case 

 was given by Mr. de Moivre. 



Mr. Bernoulli's addition, &c. can be applied to serieses of this kind. For ex- 

 ample, let the given series be — | —x A — -x 2 A- &c. = s. From this 



* ' & e e + 1 ' e + 2 



series subtract the same series diminished by m terms, viz. x m A 



' e + m e + »» + 1 



, . 1 .,ie j,i . e + m — ex m e + m + 1 — (e + 1) X"> 



x m+i j _ — lZ .»>+2- r -&;c.andthereremains V- — - — v ^ ' — 



1 e + m + 2 e.m + e e+l.e + m+1 



e + m + 2 -(e + 2 ) JV e + . + 3-(e + 3)x- c or ^ wr . te 



* T^ e + o.e + ™ + 2 ' e + 3.e + m + 3 ' 



.... . , m — e\ e + m + 1 — (e + 1) a . e + m + 2 - (e + 2) a 

 will the series become ■ \- — — s — — — x A — — s -!— 



e.m + e e+l.e + m + 1 ' e+2.e + w + 2 



H e + m + 3 - (e + 3) a 3 I _J_ _1_ 



X + e + 3 . e + m + 3 * + UC ' e + e + 1 * + e + 2 * " ' ' * 



or" 



e + m — 1 



„ . > , az m + bz m -> + cz""- 8 + &c. 



Let the general term be z + e . z + e+1 . z + e + 2m . s + e + n _ ! X ** 



= fc^ + i+T+i + r+T+2 • • • r+rr-*--) **■ s «PP° se P = te y = r - 2 > 



£ = ^'a; 3 , . . x = x'x"-'; then will the sum of the above-mentioned series be 

 (* + P' + 7 + *' + & c.) X s - J ((3' + y + r + &c.) - j— (y' + «T'+&c.) 



1_ (j' + &c.) - &c. 



e+ 2 v ' ' 



1 X x^ 



From the sum of the series — + — &c. by these and the prin- 



ciples before delivered can be deduced the sum of any series whose general term is 



a:" 1 + bz m ~< + &c . 



2z + e. 2z + e + 1 . 2z + e -t- 2. 2z + e.+ 3 X &c. 



X X^ X^ X 



In like manner from the sum of the serieses -| — ■ -) ■— + &c. T A~ 



ee+le+2 / ' 



X s , „ i , x 2 , x 3 



1- ? 1- &c. — I ; 1 r— + &c. &c. can be deduced the sum 



462 



