JA 



'M. 



Enodatio tnstgms cttjusdam paradoxi ctrca mulliplicalionem angulorum, 313 



31. Coroll, 3. Si exponens n negative capiatur, binae sequentes series ad summani revocabuntur 



4^^ 4.8 ^^ 4.8.12 ^^ 4.8.12.16 «5 -+- etC, 



V I 



hujus serieisummaest^fi::?:^'^^"^^''^!^^^ Tum 



\ ^ J \ X J .^,, ^^^, ^ .^^^j .^^ 



\ V * 4.8 ^ 4.8.12 *'*~^ 4.8.12.16 ^—^^^> 



cujus summa est =2"( ^^*"*"''^" * Y» 



32. Problema. Hanc formulam ( ~^'^i~*~ v —^y \ Iq seriem inftnitam resolvere, cujus 

 termini secundum potestates ipsius x progredirentur. 



Solutio. Posito ^—( "^"^^"^ ^ -^)y gj.jj quadratis sumendis zz=r "*" ^^~ ) » hincque 



^=(— i — ) ' 



quae forma in priori continetur, simodo ibi loco a? et n scribatur aKC et -^? quocirca colligitur 

 statim series quaesita: 



i— "" XX 1 "^"-^) ^* n(n-8)(n-10) 6 , n(n -10)(n-12)(n- 14) « . 



* 8^^^~8J[6~^ 8.16.24 ^ "* 8.16.24.32 «J — etC. 



quippe cujus summa est' = (^ ^ -t-a;)-»-y(i-a;) y^ 



33. Coroll. Sumto n negativo, ut prodeat haec series 



I 1 "" XX X "^"-^^^ >r* ■ "(»-*-8)(»-«-*0) ^6 , n(n-»-10)(n-f-12)(n->-14) , 



,-Hg(rajH- g jg X -^ g-^g-^^^ XH 8.16.24.32 ^ -^-etc, 



• /"V(\.-t-x)-t-V(\. — x\\ — ^ 



hujus summa erit = (^— ^ -\ , quae reducitur ad hanc formam: 



/ •/(! -H a?) — /(1 — x^Y 



V 



)' 



34*. Scholion. Omnes series istas, quarum summam hic assignavi, in hac forma complecti licet: 



. n n(n-i-3) n(nH-4)(n-i-5) o n(n-*-5)(n-*-6)(n-i-7) , , 



S = 1 H V H ^ YY H ^^^ — Y H ^ — — Y •+- CtC. 



1-^ l.<i ^^ 1.2.3 ^ 1.2.3.4 ^ -T-c.,v. 



/'l_i_y(l 4m)\ — ^* 



erltque * = ( \ -) ; unde patet si fuerit h-yy^ i, seriei summam esse imaginariam; realem 



autem, si sit ky <C. i- Casu autem y = y erit, uti jam supra observavimus, 



I ♦^ n(n-t-.3) n(n-<-4)(nH-5) n(nH-5)(n-H6)(n-«-7) __ g^ 



4 4.8 4.8.12 4.8.12.16 -^ «''^- 



Verum illa series pluribus modis transformari potest, ex quibus hunc solum casum affero, qui oritur 

 differentialibus sumtis, erit sciiicet 



L. Euleii Op. posthnma. T. I. j^Q 



