VI. INFINITE SERIES 



6.00 An infinite series: 



00 



2 U n = 

 n=i 



is absolutely convergent if the series formed of the moduli of its terms: 



I A, \ I I III I I 



I 2/1 I + I ^2 I + I U2 I + . . . . 



is convergent. 



A series which is convergent, but whose moduli do not form a convergent 

 series, is conditionally convergent. 



TESTS FOR CONVERGENCE 



6.011 Comparison test. The series Sw n is absolutely convergent if | u n \ is 

 less than C \ v n where C is a number independent of n, and v n is the nth term 

 of another series which is known to be absolutely convergent. 



6.012 Cauchy's test. If 



Limit , , 

 n _^ m I u n | <i, 



the series 2u n is absolutely convergent. 



6.013 D'Alembert's test. If for all values of n greater than some fixed value, r, 



the ratio 



U n +l 



is less than p, where p is a positive number less than unity 

 and independent of n, the series ^Lu n is absolutely convergent. 



6.014 Cauchy's integral test. Let/(#) be a steadily decreasing positive function 

 such that, 



Then the positive term series Sa n is convergent if, 



is convergent. 



6.015 Raabe's test. The positive term series S# n .is convergent if, 



n( - ij^l where /> i. 

 It is divergent if, 



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