1 1 b dx x 2 b = lim b = b lim x 1 b = 1 1 b 1 lim 1 = b b = 1 (Limit exists and finite) 1 dx Hence is convergent. 1 x 2 85 9.5 Definition of absolute convergence and conditional convergence of an improper integral i).The improper integral, ( ) f x dx is said to be absolutely convergent if a ( ) f x dx is convergent. a ii).The improper integral ( ) f x dx is said to be conditionally convergent if a a b ( ) f x dx is convergent but ( ) f x dx a is not convergent. 9.6 Cauchy’s inequality for definite integrals a b b Cauchy’s inequality states that ( ) f x dx ( ) f x dx a This inequality is very useful to determine the absolutely convergent and conditionally convergent improper integrals.
b 1 1 x 2 dx = b = = lim b lim b b x 1 1 1 b = lim b 1 1 b = 1 (Limit exists and finite) Hence 1 1 x 2 dx is convergent. 85 9.5 Definition of absolute convergence and conditional convergence of an improper integral i).The improper integral, a f ( x ) dx is said to be absolutely convergent if a f ( x ) dx is convergent. ii).The improper integral a f ( x ) dx is said to be conditionally convergent if b a f ( x ) dx is convergent but a f ( x ) dx is not convergent. 9.6 Cauchy’s inequality for definite integrals Cauchy’s inequality states that b a f ( x ) dx b a f ( x ) dx This inequality is very useful to determine the absolutely convergent and conditionally convergent improper integrals.
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