1 



510 . .0^1. EULERI OPEIU POSTHLMA. Anaiysis. 





cujus posterioris menibri integrale est — ~ (^l {i -i- x)f— — ~ {12)^. \^^ 



2 \'v- --// 2 



rdx 



Pro primo merabro /— l{l-h-x), id erit 



/dx C XX x"^ a;* aj^ . \ 



XX x'^ x^ 



Unde facto a7=l, erit haec pars > Y'^* 



^ 1 1 1 . n^.T 



1-4-^7 -16-^ <="=•= 12 



)1 itid 



^TT 1 



Consequenter habebimus < = 7^ g ('^S) , ergo summa seriei propositae 



12 



*=i-^i<'2)' 



etc. 



Theobema. Sequentis seriei 



summa erit 5 = -^-Q \ 



Demonstratio. Colligantur hic iterum termini postremi singulorum membrorum: . 



•••-• : , iiuSrjonpaii cborff ; 



.111 . TCTt 



*-»-T2-^p-^7-^-^ ^*^- =-8-* 

 His deletis reliquorum ultimi termini colligantur, qui sunt /2 



J L J , _ ± 1 '• ^ ,7=-^*n' 



1.3 3.5 — 5.7 ^^^" — ~" 2 -^y ^ ^ 



11 1/lN 



Sequentium ultimi dant ~*~ 7~5 ~*~ 3^^ -*- etc. =-4 (^"+"3") 5 sequentes erunt 



^'^W rn«)Hic oci_ ^ . .^ ■mua sbar 



1 1 1 1 /. 1 1\ 



-■" 1.7 ■"3.9"' 501" — " 6"V"^3 -^ 5; V. 



- := .,.4. f ■ ■ ■ .... ' •■ -_ J:' 



; jCjt 1 1 / 1 \ 1 / 1 1 \ 



et ita porro. Hinc erit S = -g t, existente « = — .1 — — M-f-_j-i-_MH-— -h— j— etc. Statuatur 



,=:_..l-_ _.(^l-H-3-)+ g-^l-H-g-H^-g)- etc. fietque 



^ = ^-jr3(^l+i-)-i-«.^(l-H-l -h4)- etc. 

 cujiis seriei primi termini collecti dant x — x^ -\- x^ — x' ~\~ etc. = ^j • Secundi termini : 



x^ x^ x^ 1 a? 



nr-t — r 7^-^ etc. = — -r 



3 1 -i-aMr' 



1 a^ . 



sequentes dabunt h- -=- • 5 > sicque erit 



a; — - a?'* -f- -- a;* — — a;' -1- etc. 



"* o D 7 Arc. tang x 



dx 1 -H XX 1 -^ XX 



/dx /* 'dx 'l 



i • 1-, » cujus integrale t = — (Arc tang x)^. Hinc sumto a- = 1, erit 



^ ^ 1 fl^^ ^T-^r - ftrt jtn ZftTt 

 '= 2 -16 = 32' ^«"«equenter 5 = -^ - -g^ = -g^- 



