SERIES. 



5- Po'wers of Arilhmetkals. 

 {m+pY + (m + 2p)' + {m + ^p)'' . . . . {m + np)' 



+ _L >-(r-i) (r-2) (r-3) (r - 4) _, _ _l_ r(r-i) (r-2) . . . (r - 6) . 



6 2.3.4.5.6.7 ^ 30 2.3....8 



I r (r- l) (r-2) ... . (r~8) 



+ ^ — i '- /.' n'--" - &c. 



42 2 . 3 .... 9 



?!!: _ "f"'" . r(r-l) (r-2) .„,_.,_ 



+ — ^^ '—^ 'p'^m'—'^— &C. 



(r-fl)/. 2 3.4 2.3.4.5.6 



continued till they terminate. The co-efficients are the fame as - , -t^-, Ike. N'' 11. 



A 13 



This form includes all the powers of the natural feries, of which we have given the particular forms to the fifth power, 

 under art. 3. 



6. Series ofjigurate Numbers. 



I + m + !!Lif!±l} 4. " (>" + l) {»' + i ) m(m + l) (m + 2) (m + 3) ^^ 



1.2 1.2-3 1.2.3.4 



-g _ n{n+ l) [n+ 2) (« + 3) to m terms, 

 1.2.3.4 torn terms. 



See particular refults, art. 3. 



7. Series of compound jlrithmeticals. 



{m + e) ip + e) + (m + 2e) {p + 2e) + . . . . {m +■ ne) {p + ne) 



(S) =nmp+"-^' (^ + ^), + L "+0(^" + i .)^, 



8. Series of compound geometrical Numbers. 

 {b - m) (c -p) e^ + {b - 2 ,n) (r - 2/) f'*' + (b - ^ m) (c - ^p) f^" + &c. 



(s) = -il- j i, _ ^^lif + iHi+U I 

 ' I - ;- 1 I - (-^ ^ ( I - f -^j ' j 



9. Series of compound Jigurate Numbers. 



I + mx + l(^Jli) ^. , '"(w + l) {m + 2) _^, ffi (m+ 1) (m+ 2) (m-f 3) 

 1.2 1.2-3 1-2-3-4 



10. Series of the Reciprocals ofjigurale Numbers. 



^ I . 2 1.2.3 I-2-3.4 



m OT (m + l) m (ot -f i) (m — 2) m (m + I) (m + 2) (ot + 3) ''" 



(.) = ^H-' 

 m — 2 



II. Reciprocals of the Powers of Arilhmeticals. 



1,1 1 I I 



a" (a±^)" (a+2rf)'" ^ ('»±3'^)"' ' ' ' {a±{n — \)dY" 



^' {m—l)a'"-'d za" 2.A.a"'+' 2 . 3 . 4 . . . B a'" + ^ 



, (^"-^'oai) w(>H + i) (m + 2) . . . (m + 4 )^' 

 "^ , 2 . 3 . 4 . 5 . 6 . . . C «'« + s *'^' 





where 



