XIV USEFUL FORMULAS. 



e. The binomial series and applications. 

 For a > b, 



{a ± by = rt" ± ;/ ^"-1 ^ -f ^LkLz^ ^n--2 ifi 



n (u — i) (n — 2) 

 ± X...3 ^-a'^-^b^+.... 



For X < I, 



/ . \« , I '^ ('^ ~ o , " (« — i) (n — 2) „ , 



(i ± xy = i±nx + \^2 •^' ± ~^ — TT^H — ' -^ + 



* o 



J I ^y = I — a: + a-- — ^3 _j_ ^4 _ _ ^ 



(I + ^^0^ = I + i -^ - i ^^-' + tV^^-^ - Tiff :^^+ . . . 

 (i - a-)^ = I - i ^ - I x'' -^i^x^- T-ij ^* - . . 



^^3p^ = I _ J a- + § a-2 - T^ X3_|_ ^3^5^ ^^.4 _ _ _ 



(I - xf 



f. Exponential and logarithmic series. 

 For — CO < a: < 00, 



^^=1+- + — +-^ + — ^— + .... 



' I ' 1.2 ' 1.2. 3 ' 1.2.3.4 I 



The number e is the base of the natural or " Napierian " system of logarithms. 

 For a- = -|- I, the above series gives 



^= 2.718281828459 .... 

 In the natural system the following series hold with the limitations indicated : 



I log ^ I (log ^y 2 I 0°'- of 



' I ' 1.2 '1.2.3 



CO < .r < CO ; 



x" .Y* JC'* X^ 



log (i-\-x) = x- ~ -\-~ - Y + y- 



X ^ I \ 



log (i — jc) = — :r — 



jy^ -Y^o ^4 'V* 



*.T/ *.^ t-T' vv 



2 3 



X < I ; 



'°-=^{^;-^+K:-+-^)+K.^)+K^+-;)'+- } 



o < a- < 00 ; 



y- < (2 a- 4- vf. 



