Question

In: Math

Real analysis

The Couchy product of two absolutely convergent series converges absolutely.

Solutions

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 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 converges,

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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