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Unformatted text preview: c Gabriel Nagy The StolzCesaro Theorem The Theorem. If ( b n ) n =1 is a sequence of positive real numbers, such that n =1 b n = , then for any sequence ( a n ) n =1 R one has the inequalities: lim sup n a 1 + a 2 + + a n b 1 + b 2 + + b n lim sup n a n b n ; (1) lim inf n a 1 + a 2 + + a n b 1 + b 2 + + b n lim inf n a n b n . (2) In particular, if the sequence ( a n /b n ) n =1 has a limit, then lim n a 1 + a 2 + + a n b 1 + b 2 + + b n = lim n a n b n . Proof. . It is quite clear that we only need to prove (1), since the other inequality follows by replacing a n with a n . The inequality (1) is trivial, if the righthand side is + . Assume then that the quantity L = lim sup n ( a n /b n ) is either finite or , and let us fix for the moment some number ` > L . By the definition of lim sup, there exists some index k N , such that a n b n `, n > k. (3) Using (3) we get the inequalities a 1 + a 2 + + a n a 1 + + a k + ` ( b k +1 + b k +2 + . . . b n ) , n > k. (4) If we denote for simplicity the sums a 1 + + a n by A n and b 1 + + b n by B n , the above inequality reads: A n A k + ` ( B n B k ) , n > k, so dividing by B n we get A n B n ` + A k `B k B n . (5) Since B n , by fixing k and taking lim sup in (5), we get lim sup...
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This note was uploaded on 11/03/2011 for the course MATH 1090 taught by Professor Greenwood during the Spring '08 term at MIT.
 Spring '08
 greenwood
 Calculus, Real Numbers, Inequalities

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