for the Valuation of Life-annuities, 
125 
and a x a—A(—A x a— — —* 
s ' a* 
Therefore, which is the firft theorem. 
Again, Ax«=i+^+p> See. to^> 
and bx«=i+j+£ + £> &c. to ~~ * 
Therefore, bx (2- ax #=-+4 + 4> Sec. to • 
To both tides of the laft equation add- > and it will ap- 
pear, that 
n 1 
n— I n 
bx a- ax a +— - 3 + 4+ ’ Bee. to -- — - + — — b . 
a* a ar a* ar a "— 1 a n 
A x a 
Therefore, b xa — b = b x a — irAx« — — ; andB — ' 
a n a — 1 a n + l —a n 
For a, in this laft equation, fubftitute its equal, or 
~~ x ^T~i 9 and t ^ ie refulting equation will be 
===71= b, which is the fecond theorem* 
a — x ) an a— l a » 7 
When n is infinite, all but the firft terms in both 
thefe theorems vanifh; and therefore, —— is the fum of 
the feries - + \ +4 ’ Sec. continued infinitely ; and ==& is 
a a a y a — 1| 
id? 
the fum of the feries - +jr+^r’ 8tc. continued infinitely. 
By 
I 
