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In: Math

Real analysis

The Couchy product of two absolutely convergent series converges absolutely.

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Expert Solution

Let bn. рир “р  be two absolutely convergent series. Then the sequence

                                  Sn ΣΣ|a,bj| converges. 1=1=1

Suppose Σα| = Q and and Σα, = β

then,

                                 Sn Σ|a:ΣΙ6;l < αβ 1=1 =1

and {sn  being monotone sequence that is increasing and bounded by above must converge.

Define, Cn -Σαμβn-k- k=1

Then Сп  Converges absolutely. we have,

                 η n ca| = Σαρδη- <Σαβη- -k k=1 k=1

and 

             ΣIcal <ΣΣΙa,θε-; <ΣΣΙa;θι k=1 k=1 i=1 i=1 k=1

This shows Σ |cal S 5η  and since Sn  converges,

Сп  must also converge absolutely.

Hence, The Couchy product of two absolutely convergent series converges absolutely.

Let bn. рир “р  be two absolutely convergent series. Then the sequence

                                  Sn ΣΣ|a,bj| converges. 1=1=1

rahul answered 3 years ago
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