here, observing that 
we have 
and hence 
n (n + 1) (n + 2) — (n — 1) n (n + 1) = n (n + 1) (n + 2 — n - 1), = 3n (n + 1), 
Vn+^in {n + 1) (n + 2); 
1 4- 3 + 6 + ... + \n (n + 1) = (n+ 1) (n+ 2), 
as may be at once verified for any particular value of n. 
Similarly, when the general term is a factorial of the order r, we have 
r+1 n(n+ 1) ...(n + r — 1) n (n + 1)-. (n + r) 
+ 1 + ”‘ + — 172 7^ r " 172 ... (r + 1)' 
5. If the general term u n be any rational and integral function of n, we have 
n A n(n— 1) Ao n (n — 1) ... (n — p + 1) A 
u n = u 0 + - Alio + \ q AX + ... + — — A^i 0 , 
1 1.2 1.2...p 
where the series is continued only up to the term depending on p, the degree of the 
function u n , for all the subsequent terms vanish. The series is thus decomposed into 
a set of series which have each a factorial for the general term, and which can be 
summed by the last formula; thus we obtain 
/ \ ( n + 1) n a ( n + 1) n (n — 1)... (n — p + 1) . 
u 0 + u x ... + u n — (?i -f 1) Ub H 1 a ■ ■■ A.u 0 + ——¿r—^ ——r-r A p u 0 , 
1.2.3 ... (_p + 1) 
which is a function of the degree p + 1. 
Thus for the before-mentioned series 1 + 24-4 + 8 + ..., if it be assumed that the 
general term u n is a cubic function of n, and writing down the given terms and 
forming the differences, 1, 2, 4, 8; 1, 2, 4; 1, 2; 1, we have 
, n n(n — 1) n(n — l)(n — 2) ( , . „ 
u n = 1 + :r 4 ^ cx ' H ^—ir - ?; {= i ( n + 5n + 6), as above}; 
1.2 
1.2.3 
and the sum 
0 + 1)71 Oi + l)w(?i-l) (n + l)w(w-l)(rc-2) 
u 0 + u l + ...+u n -n +1+ 1> "2“-+ 1^273 + 172.3.4 » 
= J4 (n 4 + 2?i 3 + llw 2 + 34n + 24). 
As particular cases we have expressions for the sums of the powers of the natural 
numbers— 
l 2 + 2 2 + ... + v? = ^n (n + 1) (2n + 1); l 3 + 2 3 + ... + n 3 = ln 2 (n + l) 2 : 
observe that the latter = (1 + 2 ... + n) 2 ; and so on. 
6. We may, from the expression for the sum of the geometric series, obtain by 
differentiation other results: thus 
1 + r + r 2 + ... + r n ~ 1 = — 
1 — r 
78—2