Pagina:Werkecarlf03gausrich.djvu/155

esse positivam. Sit $h$ integer talis, ut $h+b-B$ fiat quantitas positiva, statuamusque

$M\left({\frac {m}{m-1}}\right)^{h}=N,\quad M^{\prime }\left({\frac {m+1}{m}}\right)^{h}=N^{\prime },\quad M^{\prime \prime }\left({\frac {m+2}{m+1}}\right)^{h}=N^{\prime \prime }$ etc.

atque $(tt-1)^{h}P=Q,$ $t^{2h}p=q,$ ita ut ${\tfrac {N^{\prime }}{N}},$ ${\tfrac {N^{\prime \prime }}{N^{\prime }}}$ etc. sint valores fractionis ${\tfrac {Q}{q}}$ ponendo $t=m,$ $t=m+1$ etc. Quum itaque habeatur

$Q=t^{\lambda +2h}+At^{\lambda +2h-1}+(B-h)t^{\lambda +2h-2}$ etc.

$q=t^{\lambda +2h}+At^{\lambda +2h-1}+bt^{\lambda +2h-2}$ etc.

atque per hyp. $B-h-b$ sit quantitas negativa, termini seriei $N,$ $N^{\prime },$ $N^{\prime \prime },$ $N^{\prime \prime \prime }$ etc. post certum saltem intervallum continuo decrescent, adeoque termini seriei $M,$ $M^{\prime },$ $M^{\prime \prime },$ $M^{\prime \prime \prime }$ etc., qui illis resp. semper sunt minores, dum continuo crescunt, tantummodo versus limitem finitum convergere possunt. Q.E.P.

Si termini seriei $M,$ $M^{\prime },$ $M^{\prime \prime },$ $M^{\prime \prime \prime }$ etc. post certum intervallum continuo decrescunt, accipere oportet pro $h$ integrum talem, ut $h+B-b$ sit quantitas positiva, evinceturque per ratiocinia prorsus similia, terminos seriei

$M\left({\frac {m-1}{m}}\right)^{h},\quad M^{\prime }\left({\frac {m}{m+1}}\right)^{h},\quad M^{\prime \prime }\left({\frac {m+1}{m+2}}\right)^{h}$ etc.

post certum intervallum continuo crescere, unde termini seriei $M,$ $M^{\prime },$ $M^{\prime \prime }$ etc., qui illis resp. semper sunt maiores, dum continuo decrescunt, necessario tantummodo versus limitem finitum decrescere possunt. Q.E.S.

IV. Denique quod attinet ad summam seriei, cuius termini sunt $M,$ $M^{\prime },$ $M^{\prime \prime },$ $M^{\prime \prime \prime }$ etc., in casu eo, ubi hi in infinitum decrescunt, supponamus primo, $A-a$ cadere inter $0$ et $-1$, sive $A+1-a$ esse vel quantitatem positivam vel $=0.$ Sit $h$ integer positivus, arbitrarius in casu eo, ubi $A+1-a$ est quantitas positiva, vel talis qui reddat quantitatem $h+m+A+B-b$ positivam in casu eo ubi $A+1-a=0.$ Tunc erit

$P(t-(m+h-1))$	$=t^{\lambda +1}+(A+1-m-h)t^{\lambda }+(B-A(m+h-1))t^{\lambda -1}$ etc.
$p(t-(m+h))$	$=t^{\lambda +1}+(a-m-h)t^{\lambda }+(b-a(m+h))t^{\lambda -1}$ etc.

ubi vel $A+1-m-h-(a-m-h)$ erit quantitas positiva, vel, si haec fit $=0,$ saltem $B-A(m+h-1)-(b-a(m+h))$ positiva erit. Hinc (per I) pro quantitate $t$ assignari poterit valor aliquis $l,$ quem simulac transgressa est, valores fractionis ${\tfrac {P(t-(m+h-1))}{p(t-(m+h))}}$ semper fient positivi atque unitate maiores. Sit $n$ integer