HO MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 



6.020 Alternating series. A series of real terms, alternately positive and nega- 

 tive, is convergent if a n+ i^a n and 



limit 



a n = o. 

 n oo 



In such a series the sum of the first s terms differs from the sum of the series by 

 a quantity less than the numerical value of the (s + i)st term. 



limit 



6.025 If 



= i, the series 2u n will be absolutely convergent if 



there is a positive number c, independent of n, such that, 



Un+l 



limit 

 n 



-I = -I - C. 



6.030 The sum of an absolutely convergent series is not affected by changing 

 the order in which the terms occur. 



6.031 Two absolutely convergent series, 



S = ui + uz + u 3 + ..... 

 T = vi + % + % + ..... 



may be multiplied together, and the sum of the products of their terms, written 

 in any order, is ST, 



ST = uiVi + UzVi + MM + ..... 



6.032 An absolutely convergent power series may be differentiated or inte- 

 grated term by term and the resulting series will be absolutely convergent and 

 equal to the differential or integral of the sum of the given series. 



6.040 Uniform Convergence. An infinite series of functions of x, 

 S(x) = ui(x) + u*(x) + ut(x) + ...... 



is uniformly convergent within a certain region of the variable x if a finite number, 

 N, can be found such that fo"r all values oin^N the absolute value of the remain- 

 der, | R n | after n terms is less than an assigned arbitrary small quantity e at 

 all points within the given range. 

 Example. The series, 



is absolutely convergent for all real values of x. Its sum is i + x 2 if x is not zero. 

 If x is zero the sum is zero. The series is non-uniformly convergent in the neigh- 

 borhood of x = o. 



