INFINITE SERIES TII 



6.041 A uniformly convergent series is not necessarily absolutely convergent, 

 nor is an absolutely convergent series necessarily uniformly convergent. 



6.042 A sufficient, though not necessary, test for uniform convergence is as 

 follows : 



If for all values of x within a certain region the moduli of the terms of the 

 series, 



are less than the corresponding terms of a convergent series of positive terms, 



where M is independent of x, then the series S is uniformly convergent in the 

 given region. 



6.043 A power series is uniformly convergent at all points within its circle of 

 convergence. 



6.044 A uniformly convergent series, 



S = ui(x) + uz(x) + ..... 

 may be integrated term by term, and, 



6.045 A uniformly convergent series, 



S = ui(x) 



may be differentiated term by term, and if the resulting series is uniformly 

 convergent, 



CO 



d ' d 



6.100 Taylor's theorem. 



/(* + H) = /(*) + f(x) + /"(*) + .... + 



6.101 Lagrange's form for the remainder: 



6.102 Cauchy's form for the remainder: 



ljn 



+ Ok) - 



