Chapter 7: Q. 11 (page 603)

Give examples of sequences satisfying the given conditions or explain why such an example cannot exist.
Two convergent sequences $\{a_{k}\}, \{b_{k}\}$ such that the sequence $\{\frac{a_{k}}{b_{k}}\}$ diverges.

Short Answer

Expert verified

Examples satisfying the given conditions is $\{a_{k}\} = \{\frac{1}{k}\}, \{b_{k}\} = \{\frac{1}{k^{2}}\}$ .

Step by step solution

Step 1. Given information.

Consider the given question,

Two convergent sequences are $\{a_{k}\}, \{b_{k}\}$ .

Step 2. Find if the sequences converges.

Consider the sequence $\{a_{k}\} = \{\frac{1}{k}\}$ .

The sequence is strictly decreasing and is bounded.

Therefore, the sequence is convergent and converges to $0$ .

Consider the sequence role="math" localid="1649176751776" $\{b_{k}\} = \{\frac{1}{k^{2}}\}$ .

The sequence is strictly decreasing and is bounded.

Therefore, the sequence is convergent and converges to $0$ .

Step 3. Find if the sequence satisfies the conditions.

The sequence $\{\frac{a_{k}}{b_{k}}\} = \{k\}$ is increasing sequence and is not bounded above.

The monotonically increasing sequence which is bounded above is convergent.

The sequence $\{\frac{a_{k}}{b_{k}}\} = \{k\}$ is increasing sequence and is not bounded above.

Therefore, the sequence is divergent.

Hence, the two convergent sequences $\{a_{k}\}, \{b_{k}\}$ such that $\{\frac{a_{k}}{b_{k}}\}$ is divergent are $\{a_{k}\} = \{\frac{1}{k}\}, \{b_{k}\} = \{\frac{1}{k^{2}}\}$

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Give examples of sequences satisfying the given conditions or explain why such an example cannot exist.
Two convergent sequences $\{a_{k}\}, \{b_{k}\}$ such that the sequence $\{\frac{a_{k}}{b_{k}}\}$ diverges.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find if the sequences converges.

Step 3. Find if the sequence satisfies the conditions.

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Give examples of sequences satisfying the given conditions or explain why such an example cannot exist.Two convergent sequencesak,bksuch that the sequenceakbkdiverges.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find if the sequences converges.

Step 3. Find if the sequence satisfies the conditions.

One App. One Place for Learning.

Most popular questions from this chapter

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Give examples of sequences satisfying the given conditions or explain why such an example cannot exist.
Two convergent sequences $\{a_{k}\}, \{b_{k}\}$ such that the sequence $\{\frac{a_{k}}{b_{k}}\}$ diverges.