Q. 84 Prove Theorem 7.24 (a). That is,... [FREE SOLUTION]

Chapter 7: Q. 84 (page 616)

Prove Theorem 7.24 (a). That is, show that if c is a real number and ${鈭�}_{k = 1}^{鈭�} a_{k}$ is a convergent series, then ${鈭�}_{k = 1}^{鈭�} c a_{k} = c {鈭�}_{k = 1}^{鈭�} a_{k}$ .

Short Answer

Expert verified

As ${鈭�}_{k = 1}^{鈭�} a_{k}$ is a convergent series, and c is constant we get c out of the summation and we prove that ${鈭�}_{k = 1}^{鈭�} c a_{k} = c {鈭�}_{k = 1}^{鈭�} a_{k}$ .

Step by step solution

Step 1. Given Information.

We are given that ${鈭�}_{k = 1}^{鈭�} a_{k}$ is a convergent series and c is a real number.

We need to show thatrole="math" localid="1652717667775" ${鈭�}_{k = 1}^{鈭�} c a_{k} = c {鈭�}_{k = 1}^{鈭�} a_{k}$ .

Step 2. Proof.

The series ${鈭�}_{k = 1}^{鈭�} c a_{k}$ can be written in expanded form as

role="math" localid="1652717611488" ${鈭�}_{k = 1}^{鈭�} c a_{k} = c a_{1} + c a_{2} + . . .$

It can be factorized and written as

${鈭�}_{k = 1}^{鈭�} c a_{k} = c a_{1} + c a_{2} + . . . {鈭�}_{k = 1}^{鈭�} c a_{k} = c (a_{1} + a_{2} + . . .) {鈭�}_{k = 1}^{鈭�} c a_{k} = c {鈭�}_{k = 1}^{鈭�} a_{k}$

Thus, for a real number cit can be shown that ${鈭�}_{k = 1}^{鈭�} c a_{k} = c {鈭�}_{k = 1}^{鈭�} a_{k}$ .

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91影视

Prove Theorem 7.24 (a). That is, show that if c is a real number and ${鈭�}_{k = 1}^{鈭�} a_{k}$ is a convergent series, then ${鈭�}_{k = 1}^{鈭�} c a_{k} = c {鈭�}_{k = 1}^{鈭�} a_{k}$ .

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Proof.

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91影视

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

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Proof.

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Theoretical and Mathematical Physics

Applied Mathematics

Statistics

Calculus

Pure Maths

Study anywhere. Anytime. Across all devices.

Prove Theorem 7.24 (a). That is, show that if c is a real number and ${鈭�}_{k = 1}^{鈭�} a_{k}$ is a convergent series, then ${鈭�}_{k = 1}^{鈭�} c a_{k} = c {鈭�}_{k = 1}^{鈭�} a_{k}$ .