24 TABLE 10.— MATHEMATICAL SERIES 



(, + ,)•= x'^^y + ^ir 1 *"^^-- 



n(n-l)...(n-m + D ^ m ym _ _ _ (y < ^ 



m ! 



v. , -. ■ n(n-\)x* , n(n-l)(n-2)* 2 

 (l±^)"=l±nx+ 2 , ± 3-j +••• + 



(„_*)!*! + ■••** <U 

 „_, _ , , n(n + l) . n(n + l)(n + 2)4r» 



C=Pl) . (n _i)!ife! + --- (J: <U 



(1 ± x )- 1 =l+x + x 2 + x 3 + x 1 + x* + ... (x 2 <\) 



(1 ± xY>= \+2x + Zx* + 4x* + Sx' + 6x* +. . . (x 2 <\) 



h 2 h" „ w , Taylor's 



f( x + h) = f(x)+hf'(x)+j ] f"(x) + ...+ _/«">(*) + ..-. series 



„ »- 2 r n Maclaurin's 



f(x) = f(o)+ X -i'(o) + 2\f"(° > >+--- ^\ f (o)+ --- series 



*= lim ( 1 +i)" = 1+ r! + 2i + 3T + 4I +••• 



1 , i , (jrlogq) 2 , (jrloga)' , (f^CK)) 



a'= 1 4-;rloga + 2] ' 3l f"-"- (,jr<^w; 



■i-« i 7 1 + K £ f 1 )' + K £ T 1 )'+;" 



= (4: _l)_i(^-l) 2 + H^-l) 3 ---- (2>*>0) 



-»[^ + i(ffi) ,+ i(^)' + --] 



log (1 +*) = -*•- i * 2 + 4 * 3 -i* 4 +--- ( * S< 1} 



C0S *=I (*«• + *-«') =l-f^ + fj-^ +-'- =1 -versing (* 2 <00) 



, x 3 , 2* 5 , 17* 7 ,62 , 

 tan^=^ + ^ + 1I + ^ I 5-+2835^ + ••• 



sln *=2 _COS * = * + 6~ + 2 4 T + I'4 6 7 + "- U<U 



tan- 1 A:=f-cot- 1 x = ^- ^ 8 + {^-j^'+... (* 2 <D 



_-^_I+J L. (* ! >1) 



coshx = ± (e> + e-) =\ +£ + £+£ +... U 2 < 00) 



(con/inwerf) 



(*<*) 



SMITHSONIAN PHYSICAL TABLES 



