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Chapter 7: Q. 68 (page 605)

Let $\{a_{k}\}$ and $\{b_{k}\}$ be convergent sequences with $a_{k} 鈫� L$ and $b_{k} 鈫� M$ as $k 鈫� 鈭�$ and let $c$ be a constant. Prove the indicated basic limit rules from Theorem 7.11. You may wish to model your proofs on the proofs of the analogous statements from Section 1.5.
Prove that $(a_{k} + b_{k}) 鈫� L + M$ .

Short Answer

Expert verified

Hence, the theorem is proved.

Step by step solution

01

Step 1. Given Information.

The objective is to prove that $a_{k} + b_{k} 鈫� L + M$ .

02

Step 2. Forming the equations.

We use the definition of convergence for the sequence $\{a_{k}\}$ and $\{b_{k}\}$ .

The sequence $\{a_{k}\}$ converges to $L$ .

For given $蔚 > 0$ , there exists a positive integer $N$ such that

$|a_{k} - L| < \frac{蔚}{2}$ for $k 鈮� N . . . . . . . . . . . . . (1)$

The sequence $\{b_{k}\}$ converges to $M$ .

For given $蔚 > 0$ , there exists a positive integer role="math" localid="1649270148996" $P$ such that

$|b_{k} - M| < \frac{蔚}{2}$ forrole="math" localid="1649270158043" $k 鈮� P . . . . . . . . . . . . . . (2)$

03

Step 3. Proving the theorem.

Choose a positive integer $A$ such that $A = m a x (N, P)$ .

$|a_{k} + b_{k} - (L + M)| = |(a_{k} - L) + (b_{k} - M)| 鈮� |(a_{k} - L)| + |(b_{k} - M)| < \frac{蔚}{2} + \frac{蔚}{2} f o r k 鈮� P = 蔚$

Thus, for $k 鈮� P, |a_{k} + b_{k} - (L + M)| < 蔚$ and hence, $\{a_{k} + b_{k}\}$ is convergent.

Therefore, the value is $\lim_{k 鈫� 鈭�} \{a_{k} + b_{k}\} = L + M$ .

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Most popular questions from this chapter

Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.

${鈭�}_{k = 3}^{鈭�} 鈥� \frac{1}{(k 鈭� 2)^{2}}$

Use the divergence test to analyze the given series. Each answer should either be the series diverges or the divergence test fails, along with the reason for your answer.

${鈭�}_{k = 1}^{鈭�} 鈥� \frac{k!}{k^{k}}$

Find an example of a continuous function f : $[1, 鈭�) 鈫� R$ such that ${鈭�}_{1}^{鈭�} f (x) d x$ diverges and localid="1649077247585" ${鈭�}_{k = 1}^{鈭�} f (k)$ converges.

Let 0 < p < 1. Evaluate the limit $\lim_{k 鈫� 鈭�} \frac{1 / k \ln k}{1 / k^{p}}$

Explain why we cannot use a p-series with 0 < p < 1 in a limit comparison test to verify the divergence of the series ${鈭�}_{k = 2}^{鈭�} \frac{1}{k \log k}$

The contrapositive: What is the contrapositive of the implication 鈥淚f A, then B.鈥�?

Find the contrapositives of the following implications:

If a quadrilateral is a square, then it is a rectangle.

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