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Chapter 7: Q. 69 (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 such that

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

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

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

$|b_{k} - M| < \frac{蔚}{2 |L| + 1}$ for $k 鈮� P . . . . . . . . . . . (2)$

There exists a positive integer $m$ such that

$|b_{k}| 鈮� m$ for all $k . . . . . . . . . . . (3)$

03

Step 3. Proving the theorem.

Choose a positive integer $R$ such that $R = m a x \{N, P\}$

$|a_{k} b_{k} - L M| = |(a_{k} - L) b_{k} + (b_{k} - M) L| 鈮� |(a_{k} - L)| |b_{k}| + |(b_{k} - M)| |L| < \frac{蔚}{2 m} 脳 m + \frac{蔚}{2 (|L| + 1)} 脳 L f o r k 鈮� R < \frac{蔚}{2} + \frac{蔚}{2} = 蔚$

Thus, for $k 鈮� R, |a_{k} b_{k} - (L M)| < 蔚$ 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

Let ${鈭�}_{k = 1}^{鈭�} a_{k} b e a c o n v e r g e n t s e r i e s a n d {鈭�}_{k = 1}^{鈭�} b_{k} b e a d i v e r g e n t s e r i e s .$ Prove that the series ${鈭�}_{k = 1}^{鈭�} (a_{k} + b_{k})$ diverges.

Use either the divergence test or the integral test to determine whether the series in Given Exercises converge or diverge. Explain why the series meets the hypotheses of the test you select.

${鈭�}_{k = 1}^{鈭�} 鈥� k^{鈭� 3 / 2}$

Given that $a_{0} = - 3, a_{1} = 5, a_{2} = - 4, a_{3} = 2$ and ${鈭� a_{k}}_{k = 2}^{鈭�} = 7$ , find the value ofrole="math" localid="1648828282417" ${鈭� a_{k}}_{k = 1}^{鈭�}$ .

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.

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

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.

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

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