6R Mr. Jarrett on Algebraic Notation. 



Thns |« = n . (» ± >■) {n±2r) ... (n±m-\ . r), 

 ^ = n.{n-l) {71 - 2) . . . {n-m-l), 



m 



and [n = »j. (n — l) . . . 1. 



9. Let the letter S be taken as an abbreviation of the word 

 sum, and S",,, . a„ will represent the sum of n terms of a series 

 of which the wi* term is a™. 



CO , (x-l)" 



Ex. Log^.x = 5'„(-ir-'.^^^. 



10. Prop. To invert a given series, 



Sr„, .a„ = ai + 02 + a^ + &c. + a„ 



= a„ + ff„_i + a„_2 + &c. + fli, by inverting the series 



n. Prop. To divide a given series into two series. 



S",„.a,„ = (a, + flo + a-j + &c. +a,) + (ar. + , + a,+o+&c. + a„) 

 = S ' .a + S"~'. ar+„. 



12. Prop. To separate the even and odd terms of a series, 



X 



S„.a„ = fli + as + «3 + &c. 



= (a, + aa + fls + &c.) + (ao + a^ + «rt + &c.) 



00 00 



13. Theorem. If b is independent of m, then shall 



S\ba^ = i.5",„.a„. 

 For, S"„ia„. = ba^ + fta, + iff, + &c. + 6a„ 

 = 6 {«! + flo + fla + &c. + a„J 



m ' "m • 



