324 Mr. Lubbock on the Comparison of 



The sum of the series is 



+ 24. w — Wy^-^-y,\ + &^- 



Laplace has given in the 4th Volume of the 3Iecanique Cdeste, 

 p. 206, the particular value of this series which obtains when 

 the interval i which separates the values of y is indefinitely 

 diminished. 



In this case the coefficient of A'l/^ - A«j/„ is found by inte- 

 grating 



i. (i— \) (i — 2) (i — q)di 



1.2.3 q + 1 ' 



from i=o to i=n, if n=i, the sum of the values of 3/ or the area 

 of the curve between y„ and y,„ 



= hyo +^1 + 3/2 + iy.n 



■ -^\ ^y,n -^yo} + ~\ A>» - A^^ol • 



In applications of the former series to the calculation of 

 annuities, reversionary payments, &c. i/„„ Ay„„ /\y„„ &c. = 0. 



The first term in the series of the values of y or y,, is the 

 value of a present payment = 1, if we neglect the term 



n—l .n+l , , 



— ^2,n i^y-'-^yoi' 



and the following, and suppose the values of the annual 

 payments to be in arithmetical progression, the value of an 

 annuity on the life of a person aged 20, to commence at the 

 end of the first year. 



