Pagina:Werkecarlf03gausrich.djvu/156

maior quam $l$ simulque maior quam $h,$ sintque termini seriei $M,$ $M^{\prime },$ $M^{\prime \prime },M^{\prime \prime \prime }$ etc. qui respondent valoribus $t=m+n,$ $t=m+n+1,$ $t=m+n+2$ etc., hi $N,$ $N^{\prime },$ $N^{\prime \prime },$ $N^{\prime \prime \prime }$ etc. Erunt itaque

${\frac {(n+1-h)N^{\prime }}{(n-h){N}}},\quad {\frac {(n+2-h)N^{\prime \prime }}{(n+1-h)N^{\prime }}},\quad {\frac {(n+3-h)N^{\prime \prime \prime }}{(n+2-h)N^{\prime \prime }}}\quad$ etc.

quantitates positivae unitate maiores, unde

$N^{\prime }>{\frac {(n-h)N}{n+1-h}},\quad N^{\prime \prime }>{\frac {(n-h)N}{n+2-h}},\quad N^{\prime \prime \prime }>{\frac {(n-h)N}{n+3-h}}\quad$ etc.

adeoque summa seriei $N+N^{\prime }+N^{\prime \prime }+N^{\prime \prime \prime }+$ etc. maior summa seriei

$(n-h)N\left({\frac {1}{n-h}}+{\frac {1}{n+1-h}}+{\frac {1}{n+2-h}}+{\frac {1}{n+3-h}}+{\text{etc.}}\right)$

quotcunque termini colligantur. Sed posterior series, crescente terminorum numero in infinitum, omnes limites egreditur, quum summa seriei $1+{\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{4}}+{\text{ etc.}}$ quam infinitam esse constat etiam infinita maneat, si ab initio termini $1+{\tfrac {1}{2}}+{\tfrac {1}{3}}+{\text{etc.}}+{\tfrac {1}{n-1-h}}$ rescindantur. Quare summa seriei $N+N^{\prime }+N^{\prime \prime }+N^{\prime \prime \prime }+{\text{etc.}},$ adeoque etiam summa huius $M+M^{\prime }+M^{\prime \prime }+M^{\prime \prime \prime }+$ etc., cuius pars est illa, ultra omnes limites crescit.

V. Quoties autem $A-a$ est quantitas negativa absolute maior quam unitas, summa seriei $M+M^{\prime }+M^{\prime \prime }+M^{\prime \prime \prime }+$ etc. in infinitum continuatae certo erit finita. Sit enim $h$ quantitas positiva minor quam $a-A-1,$ demonstrabiturque per ratiocinia similia, assignari posse valorem aliquem $l$ quantitatis $t,$ ultra quem fractio ${\tfrac {Pt}{p(t-h-1)}}$ semper adipiscatur valores positivos unitate minores. Quodsi iam pro $n$ accipitur numerus integer ipsis $l,$ $m,$ $h+1$ maior, terminique seriei $M,$ $M^{\prime },$ $M^{\prime \prime },$ $M^{\prime \prime \prime }$ etc., valoribus $t=n,$ $t=n+1,$ $t=n+2$ etc. respondentes, designantur per $N,$ $N^{\prime },$ $N^{\prime \prime }$ etc., erit

$N^{\prime }<{\frac {n-h-1}{n}}\cdot N,\quad N^{\prime \prime }<{\frac {(n-h-1)(n-h)}{n(n+1)}}\cdot N^{\prime }\quad$ etc.

adeoque summa seriei $N+N^{\prime }+N^{\prime \prime }+$ etc., quotcunque termini colligantur, minor producto ex $N$ in summam totidem terminorum seriei

$1+{\frac {n-h-1}{n}}+{\frac {(n-h-1)(n-h)}{n(n+1)}}+{\frac {(n-h-1)(n-h)(n-h+1)}{n(n+1)(n+2)}}$ etc.

Huius vero summa pro quolibet terminorum numero facile assignari potest; est scilicet

terminus primus

$={\tfrac {n-1}{h}}-{\tfrac {n-h-1}{h}}$