Chapter 7: Q. 84 (page 616)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given Information.

We are given that $\sum_{k = 1}^{\infty} a_{k}$ is a convergent series and c is a real number.

We need to show thatrole="math" localid="1652717667775" $\sum_{k = 1}^{\infty} c a_{k} = c \sum_{k = 1}^{\infty} a_{k}$ .

Step 2. Proof.

The series $\sum_{k = 1}^{\infty} c a_{k}$ can be written in expanded form as

role="math" localid="1652717611488" $\sum_{k = 1}^{\infty} c a_{k} = c a_{1} + c a_{2} + . . .$

It can be factorized and written as

$\sum_{k = 1}^{\infty} c a_{k} = c a_{1} + c a_{2} + . . . \sum_{k = 1}^{\infty} c a_{k} = c (a_{1} + a_{2} + . . .) \sum_{k = 1}^{\infty} c a_{k} = c \sum_{k = 1}^{\infty} a_{k}$

Thus, for a real number cit can be shown that $\sum_{k = 1}^{\infty} c a_{k} = c \sum_{k = 1}^{\infty} a_{k}$ .

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

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Proof.

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

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Proof.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

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Study anywhere. Anytime. Across all devices.

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