Summation of Series. 399 
» Tf the terms of the given feries are alternately affirmative and. 
negative, the terms of the refulting feries will be alternately 
affirmative and negative, if #2 be an odd number;  otherwife its 
terms will be all affirmative. The fum of this feries will be 
finite or infinite, as the fum of the feries r+ $ +2414 &c. is. 
finite or infinite; but from it, by the preceding method of addi- 
tion or fubtraction of Mr. BerNouLti’s, ora like method applied. 
to more feriefes, may be found the fums of different finite feriefes, 
It may be obferved,, that from Mr. BeRNouLti’s. addition or 
fubtraGiion can never be deduced the feriefes which. arife from 
this method; for, by his. method, the denominator can: never 
have any fators but what are contained in the denominators of 
the given feries, viz. (in the feries 3+ $+2+ &c.),%+/ where. 
Zis a whole number;. but by this.method are introduced into. 
the denominator the factors. 22+/, 3z+4/, &c..and 7z+/, or- 
which may be reduced to the fame (= + - ) x Me. 
If x fucceflive general terms of the feriefes arifing from Mr. . 
BeERNOULLI’s addition or fubtraCtion be added together, and in. 
the quantity thence arifing for = the diftance from the firft- 
term of the feries be fubftituted mz, there will be produced: 
feriefes of the above-mentioned. formula. 
11. Multiply two converging feriefes.a +. bx + 0x? + de? + &e.. 
=S and 4+ 0r+ 9x +&c.=V, or find any rational and inte-- 
gral function of them, and the feries. refulting will be finite- | 
and=SxV, &c. Let.e+ Cr+ 704+ &c. x"=V be finite, and. 
the refulting feries. will be finite and=5.x V, &c.. If S be a 
feries converging or not, whofe ultimate terms are lefs than, 
any finite quantity, then will the feries (¢+6x+0¢x° + &c.) x 
fa-t Px + ya" + &c.x”) = V x S be a converging one, if «+ Gx +. 
y+... cc. x"=.0; which cafe was given by Mr. De Moivre. 
Mr. 
