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Prove Theorem 7.24 (a). That is, show that if c is a real number andk=1ak is a convergent series, then k=1cak=ck=1ak.

Short Answer

Expert verified

As k=1akis a convergent series, and c is constant we get c out of the summation and we prove that k=1cak=ck=1ak.

Step by step solution

01

Step 1. Given Information.

We are given that k=1ak is a convergent series and c is a real number.

We need to show thatrole="math" localid="1652717667775" k=1cak=ck=1ak.

02

Step 2. Proof.

The series k=1cakcan be written in expanded form as

role="math" localid="1652717611488" k=1cak=ca1+ca2+...

It can be factorized and written as

k=1cak=ca1+ca2+...k=1cak=c(a1+a2+...)k=1cak=ck=1ak

Thus, for a real number cit can be shown that k=1cak=ck=1ak.

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