Q. 8 - Calculus

Question

In Exercises 4–11, give examples of sequences satisfying the given conditions or explain why such an example cannot exist.
Two divergent sequences $\left\{a_k\right\}$ and $\left\{b_k\right\}$ such that the sequence $\left\{\frac{a_k}{b_k}\right\}$ converges.

Step-by-Step Solution

Answer

The example of the sequence is $\left\{a_k\right\}=\left\{k\right\}$ and $\left\{b_k\right\}=\left\{k^2\right\}$.

Step 1. Given Information.

Two divergent sequence $\left\{a_k\right\} and \left\{b_k\right\}$.

Step 2. Consider the divergent sequence.

Consider the sequence $\left\{a_k\right\}=\left\{k\right\}$ which is strictly increasing an not bounded above.

So it is divergent sequence.

Consider the sequence $\left\{b_k\right\}=\left\{k^2\right\}$ which is strictly increasing an not bounded above.

So it is divergent sequence.

Step 3. Division of the sequence.

The sequence $\left\{\frac{a_k}{b_k}\right\}=\left\{\frac{1}{k}\right\}$ which is a decreasing sequence and bounded below.

And the sequence converges to zero.

So the division of two divergent sequence can be convergent.