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# Title: Leibniz-Handschriften zur Mathematik LH 35, 7, 7
# Folio 2r, page id 00068079:0003
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sed quoniam hoc quoque ipsum peculiarem in se habere videtur subtilitatem, et aditum aperit
ad eos casus, ubi coeffirentes potentiarum ab X sunt series repriantes, quae quiae
vel infiitae, vel finitae, sed in altioribus termmnis magis magisque crescentes, videamus an
hiae compendium sese afferat amile succedat haec methodus summam primum
ope ipsius N deinde summam provenientis verso N in n,  investigandi. Quod
sane aetificumi imprimis mivum est. Sit Ergo series y -  + +
= 2 2: Ergo gy= 4-  Mu
+1 8c
altera illa ractione assequi possimus per coninationes ne Biniones. Nempe
n+I
nntp
jam y seu = y =
si tota series ducatur in se ilsam, quilibet Terminus dhicitur in alios omnes. Unde fiet
et terminus unu in alios omnes
erit A inen et facti summa erit  in  seu verso Nin vuut jum ruo sus hoc
in quolibet instituamus, fiet: 4n-  + et et ( + debet aequari y y = 4- 8+G
na nimu
ne Ln
um
a
in incelligendo seriem numericam utcunque productam, seu indeficite finitam, et phemii
quod experimento cum provernus in hoc casu serie tantum es ad duos numeros
4m + etiam
poum producta sit y = + +  = 4 et gy - 16 videamus en s pE
est autem (aex. 4 -  e 40 4  + EF0 Videamue
8+ 4
etiam L  Est autem ( _ hic  4 + 2
1+ 1
n+ 2n +1
n+2n+
quod non procedit vs igitur acmratius investiganda. sit y = a + b + c + d. 
yy = a in a +b + CFd + b in a + b + C+ d + C in a +b+ c+ d + d in a + b+C+
e si jam a in a+bt ctd = ay et siterminus quilibet a vocetur X, fiet
gy = sxy poa, b sed jam video cur res non procedat, nam foret
Exy = a ina + b in a+b + C in a + b+C. At yy = y sx = gy nam sx - y
Alia igitur ratiine in compendia inquirendum.
yy - (xy = a in b+ ctd + bin ctd + c in d
sit y- 1 12 +3+4 +X fol y 1, Est atem jam a in b+ c+ d et bin etd
et C in d’exprimi potest, si cogitemus seriem inverso modo sumtam daci in terminum
enci semper in terminum proxime majorem, nempe ultimus seriei ordinalis
sit N, serieique inverso modo expressae prodibit natura, si ubique pro n
scribamus Y = N- n et siet n = N - r. quo et terminus seriei )
numerando scilicet contrart Erit (Y) ponendo ( r esse ordinatem contrarium
sit ergo (n) n sit exempli gratia series quadratorum 1 + 4 + 9 + 16 = y.
cujus quilibet Terminus (n) et L ∞, 4, 4sit1 vel 1 vel 3ec sit n. quaeritur
et secundo: 1 in 4+9+6 + 4 in 9+ 16 +9 in 16 = ((r + 1) in S(r) =  in Y y
e
Et ita possumus retinerus (N) ergo yy = ((2) is +) (2) in Y- y. Sed hoc
identicum est, nam nam S(n) in Y -y = ((2) Y - ((2) y et (()Y = Y s(n) = yy
unde fit yy = 5(yy + yy = ()y. At si retineamus Y videamus quio
prodeat:  quia (1) = n fit et n = N- Y vel R  X (nam N et R coinciduat)
Xr1 Narr
fiet; (Y) = N2Nrar N - 2 Ar = rr
Egps y = (n fiet yy = fri fi + ( rrrr in ) sN acum = fr
n+na
videamus an
2
Quod est plane mitabile ratociundi genus, et fier genemliter yy= f(Y per
e
sit unus ex terminis  NNN
e
endastimutes
ex
2
In
AEN
Generaliter igtur sit y = ((n) et r- s fiet   = Sen) sin) + s rn) pr
ubi tantum (1) eodem modo ontur ex _ I ex H ex N- Y  (vel (Y+ 1) ex  N- r. 1)
 ut (n) oritur ex n
Ergo si y = sSn fier yy S n su +
(
1+
p
-ENY +rr in (N- 2NrE
+Z Y
Generaliterigitur si y fn ft y = S() et r = N - n fier y = sn = ((2) () + sr+ 1) Sr
Eodem modo licelit, tale theorema comminisci pro cubo, et pro alia potentia quavis
pu si. 0 e ar pterit
2n +ED: 2
1+19
AA
norga
+IMFI
sn