S E R
this feries will be T' =:
rp 4-ZZ 4'Z + I rp
■1X22 — I
T *?,
2ZX 2Z -J- I
where the fucceflive values
Mr. SftrUtigs ?netbodus diffe-
4224-22; t>f zare 1, 2, 3, 4, Szc. S© rentialis, in the introduction. This may fuffice to convey a notion of thefe differential equa- tions, defining the nature of 'feries' 's. But as to the applica- tion of thefe equations in finding the fums of feries' 's, it would require a treatife to explain it. We muff, therefore refer the curious to that excellent one juft quoted, as alfo to Mr. de Moivre's Mifcellanea Analytical and feveral curious papers by Mr. Euler in the a£ta Petropolitana.
A feries often converges fo flowly as to be of no ufe in practice. Thus if it were required to find the fum of the feries — J.
■ h r& + J7s + gTT^ + ^ c * Wnicn lora " Brounker
found for the quadrature of the hyperbola true to 9 figures, by the mere addition of the terms of the feries, Mr, Stirling computes (hat it would be neceffary to add a thoufand mil- lions of terms for that purpofe ; for which the life of man would be too fhort. But by that gentleman's method the fum of the feries may be found by a very moderate computa- tion. See Meth. Diff. p. 26, feq. Recurring Series, is ufed for a feries which is fo conftitut- ed, that having taken at pleafure any number of its terms, each following term fhall be related to the fame number of preceding terms according to a conftant law of relation. Thus in the following _/«■/«, ABC D E F
1 4- 2 x + 3 xx 4. 10 *J + 34 ** 4. 97 x s 4. &c, in which the-terms being refpectivcly reprefented by the ca- pitals A, B, C, D, &c. we lhall have
D = 3 C * — 2 B ** -f- 5 A* J E=3D^- 2 C ** + 5 B * 3 F = 3 E * — 2 D ** -[- 5 C * 3 &c; Where it is evident, that the law of relation between D and E is the fame as between E and F, each being formed in the fame manner from the three terms which precede it in the fertes.
The quantities 3 * — 2 xx 4- 5 # 3 taken together and con- nected with their proper figns is what Mr. de Moivre calls the Index or the fcale of relation. Sometimes the bare coeffici- ents 3 — 2 -f- 5 are called the feale of relation. And this ■ fcale of relation fubtracred from unity is called the differential fcale. Thus in the foregoing ym« the fcale of relation be- ing 3 at — 2 x x 4- 5 x 3 , the differential fcale will be 1
3 X 4" IX X "5 A' 3 .
On the fubject of recurring feries fee Mr. de Moivre's Mif- cellan. Analytic, p. 27. feq. and p. 72. feq. as alfo his doc- trine of chances 2d Edit. p. 193. feq. and Mr. Euler's analyf. infinit. Tom. I. p. 175. feq.
A recurring feries, with its fcale of relation being given, the fum of that feries continued in infinitum -will alfo be given. For mftance, fuppofe a feries a-}~l>x-l-cxx4- d x 3 4- e x* 4- & c - where the relation of the coefficient of any term to the coefficients of any two preceding terms may be expreffed hyf—g; that is, e=fd — ge ; and ^ = f c — gb Sec. Then will the fum of the feries continued in in- finitum be^ a 4- h x
—fa x
! ~fx + gXX
To facilitate the intelligence of this rule by a particular ex- ample, affume any feries whereof the two firft coefficients are given, fuch as 2 and 5, and fuppofe/ and g to be refpec- tively 2 and 1 ; then we (hall have the following feries
2 4- 5 x 4- 8 x x 4- 1 1 x 3 -f- 14 a 4 4- 18 a- 5 4- &c.
And the fum — — — ■= 1 — — — , ! _ .
1 — 2 * 4- x x jzr~ x z
For the proof divide I by 1 — *
1 — 2*4-**) 1 (1 4-2*4-3**4-4 * 3 4- &c. which multiplied by 2 4. x
2+4*4-6**4-8 x 3 4- &c.
4- * 4- 2 xx 4- 3 x 3 4. & c.
2 -f-5*+8**-]-ji* 3 4- &£ = the giv-
gives
the product =r en feries.
Analogous rules might be derived for more complex cafes ; and Mr. de Moivre's general rule is,
i 8 Take as many Terms of the feries as there are parts in the fcale of relation. 2°. Subtract the fcale of relation from unity, the remainder is the differential fcale. 3 . Multiply the terms taken in the fries by the differential fcale, begin- ning at unity, and fo proceeding orderly, remembring to leave out what would naturally be extended beyond the laft of the terms taken. Then will the product be the numera- tor, and the differential fcale will be the denominator of a fraction cxprelTing the fum required.
But we muft here obferve that when the fum of a recurring feries extended ad infinitums found by Mr. de Moivre's rule, it ought to be fuppofed, that the feries converges indefinitely, that is, that the terms may become lefs than any aifigned
S E R
quantity. For, if the feries diverges, that is, if its terras con- tinually increafe, it is not true, that the rule gives the fum. For the fum in fuch cafes is infinite, or greater than any given quantity, whereas the fum exhibited by the rule will often be finite. The rule therefore in this cafe only gives a fraction, the reduction of which into a feries gives the pro-
pofed feries. Thus ==- reduced to an infinite feries »ives
1 — x °
the recurring feries i ■+- 2 x + 3 xx + &c. whofe fcale of relation is 2 — 1 , and its fum by the rule will be
a + b x — fa x 1 + 2 x — 2 x _ I
1 — fx -f- 1 x x Y—2x + itx ~~ frf>— the
quantity from which the feries arofe. But this quantity can- not in all cafes be deemed equal to the infinite feries 1 4- 2
- + 3 * * &c - For flop where you will, there will always be
a fupplement required to make the produfl of the quotient by the divifor equal to the dividend. When the feries, indeed, converges infinitely, the fupplement diminifhing continually) it becomes lefs than any afligned quantity, and is therefore without error, reputed nothing; but in diverging fries this fupplement becomes indefinitely great, and the fries deviates indefinitely from the truth. See Mr. Ceipris comment on Sir If. Newtm's method of fluxions and infinite fries p 1 e 2 Mr. Stirling's method. Differentialis p. 36. Bernoulli de Serieb. infinit. p. 249. Cramer, Analyfe des lignes Cour'bes p. 174.
A recurring feries being given, the fum of any number of the terms of Aim feries may be found. This is prob in p. 73. Mifcel. Analyf. and prop. V. p. 196. of the Docirine of chances. One folution of the fimpleft cafe will be fuffici- ent to give an idea of the method there ufed. Let there be a geometric progrefiion a -f- a x + a x* -f a x l + &c : it is required to find the fum of a number n of its terms. 1 hen will the laft term bcsi.-.. Suppofe the progreffion continued admfiritum, and we mail have two in finite progrejlions, the firft beginning with a and the fecond with ax: Now the difference of the fum of thefe two muft be the fum ol the number n of terms. By the rule the fum of the firft infinite progreffion will be — - ; and the
I — - X
fum of the fecond will be — — . But g r ax*
a — ax« 1 ~ X I ~* i= ~ x ~
— j-~> which will therefore be the fum of a number n, of
terms.
I his quantity is equal to ■
-' which laft ex-
preffion, calling **»—=/, w m be equivalent to this, -j— - » which is the common rule for finding the furn of
any geometric progreffion of which a, the firft term • ratio ; and I the laft term are given.
the
The feries refulting from the divifion of unity by T~"7. f or ■= 1 — x is 1 4- p x -I- 1 x LjlJ 1
P ~ ^ P ~~ *' + & c - And the fum of any number
of terms expreffed by n of this feries will be
n+i
TZTf
n+l
&c. which is to be continued till the number of terms == p. See Mifcel. Analyf. pag. 167. Docirine of chances, p. 196. This theorem is of u(e in finding the fums of progreffions of figurate numbers, and others.
Suppofe, for inftance, it were required to find the fum of any number n of terms of the geometric progreffion I -f- * +
x x -\- xs -f- &c. generated by -
Here^ =: 1. And
the fum will confequently be = — 1 — •
Again, if the fum of a number n of terms of the feries l + 2»4-3«-(-t* I + !ic ' were required. The_/in'«
is generated from _, Then p =s 2. And the fum ra
" — x
I then will the fum be
n x" _ — • Suppofe
«+i
which is the fum of the arithmeti-
cal progreffion 1 + 2 + 3 + 4 -f &c. continued to the number n of terms. But it is to be obferved that it requires a particular artifice to derive this fum from the general rule :
for at firft fight the fum appears under this form t
