Pagina:Werkecarlf03gausrich.djvu/154

seriem $M,$ $M^{\prime },$ $M^{\prime \prime },$ $M^{\prime \prime \prime }$ etc. si non ab initio tamen post certum intervallum omnes suos terminos eodem signo affectos continuoque crescentes habituram esse.

Eadem ratione, si e differentiis $A-a,$ $B-b,$ $C-c$ etc. prima quae non evanescit est negativa, series $M,$ $M^{\prime },$ $M^{\prime \prime },$ $M^{\prime \prime \prime }$ etc. si non ab initio tamen post certum intervallum omnes suos terminos eodem signo affectos continuoque decrescentes habebit.

II. Si iam coëfficientes $A,$ $a$ sunt inaequales, termini seriei $M,$ $M^{\prime },$ $M^{\prime \prime },$ $M^{\prime \prime \prime }$ etc. ultra omnes limites sive in infinitum vel crescent vel decrescent, prout differentia $A-a$ est positiva vel negativa: hoc ita demonstramus. Si $A-a$ est quantitas positiva, accipiatur numerus integer $h$ ita, ut fiat $h(A-a)>1,$ statuaturque ${\tfrac {M^{h}}{m}}=N,$ ${\tfrac {M^{\prime h}}{m+1}}=N^{\prime },$ ${\tfrac {M^{\prime \prime h}}{m+2}}=N^{\prime \prime },$ ${\tfrac {M^{\prime \prime \prime h}}{m+3}}=N^{\prime \prime \prime }$ etc., nec non $tP^{h}=Q,$ $(t+1)p^{h}=q.$ Tunc patet, ${\tfrac {N^{\prime }}{N}},$ ${\tfrac {N^{\prime \prime }}{N^{\prime }}},$ ${\tfrac {N^{\prime \prime \prime }}{N^{\prime \prime }}}$ etc. esse valores fractionis ${\tfrac {Q}{q}}$ ponendo $t=m,$ $t=m+1,$ $t=m+2$ etc., ipsas $Q,$ $q$ vero esse functiones algebraicas formae huius

${\begin{aligned}&Q=t^{\lambda h+1}+hAt^{\lambda h}+{\text{etc. }}\\&q=t^{\lambda h+1}+(ha+1)t^{\lambda h}+{\text{etc. }}\end{aligned}}$

Quare quum per hyp. differentia $hA-(ha+1)$ sit quantitas positiva, termini seriei $N,$ $N^{\prime },$ $N^{\prime \prime },$ $N^{\prime \prime \prime }$ etc. si non ab initio tamen post certum intervallum continuo crescent (per I); hinc termini seriei $m{N},(m+1){N}^{\prime },(m+2){N}^{\prime \prime },$ $(m+3)N^{\prime \prime \prime }$ etc. necessario ultra omnes limites crescent, et proin etiam termini seriei $M,$ $M^{\prime },$ $M^{\prime \prime },$ $M^{\prime \prime }$ etc., quippe quorum potestates exponente $h$ illis sunt aequales. Q.E.P.

Si $A-a$ est quantitas negativa, accipere oportet integrum $h$ ita, ut $h(a-A)$ fiat maior quam 1, unde per ratiocinia similia termini seriei

$mM^{h},\quad (m+1)M^{\prime h},\quad (m+2)M^{\prime \prime h},\quad (m+3)M^{\prime \prime \prime h}\quad {\text{etc.}}$

post certum intervallum continuo decrescent. Quamobrem termini seriei $M^{h},$ $M^{\prime h},$ $M^{\prime \prime h}$ etc. adeoque etiam termini huius $M,$ $M^{\prime },$ $M^{\prime \prime },$ $M^{\prime \prime \prime }$ etc. necessario in infinitum decrescent. Q.E.S.

III. Si vero coëfficientes primi $A,$ $a$ sunt aequales, termini seriei $M,$ $M^{\prime },$ $M^{\prime \prime },$ $M^{\prime \prime \prime }$ etc. versus limitem finitum continuo convergent, quod ita demonstramus. Supponamus primo, terminos seriei post certum intervallum continuo crescere, sive e differentiis $B-b,$ $C-c$ etc. primam quae non evanescat