Table 7. ji 



SERIES. 



n(n-i) . . .(n-m+i) , -> ^ ■>-, 



(I ±.)» = I ±;..+ 'i^^^ ± "<^'-'\<"-""^ + . .. + ^4"%^+ •••(-^<0 

 2! 3 ! («— ^j! ife 1 



(I ± :*;)-i == IT s;+x2 T :»:'+x« T 0;^+ . . . (a;-< i) 



(I ± a;)-2 = I T 2X+3.V- T 4a;'+5:c< T (>x^+ . . . {x^<'i) 



f(x+h) - / {x)+hj' ix) + ^J"{x)+...+ J" /<«' (*) + ... '^''^'°J^ries. 



r / N r / s ^ •, . ^ ^^ ^// , ^ «" ^ , N N Maclaurin's 



/ ix) ^f{o)+-J' (0) + - /" (0) + . . . - / (") (0) + ... series. 



lim{ I+-)"=I+-:+— + — + -:+... 

 \ »/ 11 2! 3! 4I 



e' ^ i + x+~, + -, + -.+ ... (a;2<Q0) 



2! 3! 41 



..= , + ,. log „+<£i^' + f£i2£l)^ + .... («.<») 



"^ (X-I) - ^ {X - 1)2 + i (-V - IP - • • . (2>.t>0) 



log (I +A-) = :C - i .V2 + i :c3 _ 1 ^.4 + . . . . (^2<i) 



sin X = -. (e'-^ - e"'-^) = :i; - , + '--—+•.. (a;2<30) 

 21 3 • 5- 7 • 



I X^ T^ X^ 



COS a; = - (e'^ + c""'-^) = 1 , + '-;— V-. + . . . = i — versina; ix^<'x>) 



6! 



.r' 2.T* ly.r^ 62 



,a„« = . + -+--+-_ + _,. + . 



6'<t) 



sin-i a; = cos.-i x = x + z- -^ •-•- + -•-•^•—4-... (x-<i) 



2 62452467 



tan-i x = - — cot.-i .t = a; x^ -i x^ x'' + . . . (a;''<i) 



2 3 5 7 



""III / ,^ s 



=.- + —- — + ... (a:2>i) 



2 X 3j;' Sx' 



sinh a; = - (e* - c-^) = x + ~, + —, + -,+.. . {x'^<x) 



2 3'- 5' 7" 



Smithsonian Tables. 



