# Abel.pdf - REAL ANALYSIS PARTIAL SUMMATION(ABEL SUMMATION...

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REAL ANALYSIS PARTIAL SUMMATION (ABEL SUMMATION) As part of the analogy existing between summation and integration, partial summation corresponds to integration by parts. If u v and s m = m X r = u a r then we have the identity v X m = u a m b m = b v +1 s V + v X m = u s m ( b m - b m +1 ) (1) Proof v X m = u a m b m = v X m = u ( s m - s m - 1 ) b m = b v +1 s v + v X m = u s m ( b m - b m +1 ) with the convention that empty sums are zero. Abel’s lemma With the above notation, suppose that { b m } is a positive monotonic decreasing sequence, and that | s m | ≤ M for all m . Then fl fl fl fl fl v X m = u a m b m fl fl fl fl fl Mb v Proof fl fl fl fl fl v X m = u a m b m fl fl fl fl fl = fl fl fl fl fl v X m = u s m ( b m - b m +1 ) + s v b v +1 fl fl fl fl fl v X m = u | s m | ( b m - b m +1 ) + | s v | b v +1 M " v X m = u ( b m - b m +1 ) + b v +1 # = Mb 0 Theorem 6 Dirichlet’s test Suppose that φ n is a monotonic decreasing sequence converging to zero, and that a n is a series with bounded partial sums. Then X n =1 a n φ n is convergent.

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• Fall '98
• pfitz

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