Q. 6 - 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 $\{a_k.b_k\}$ converges.

Step-by-Step Solution

Answer

There is no such divergent sequence $\left\{a_k\right\} and \left\{b_k\right\}$ whose product $\{a_k.b_k\}$ is a convergent sequence.

Step 1. Given Information.

Two divergent sequences $\left\{a_k\right\}$ and $\left\{b_k\right\}$ such that the sequence $\{a_k.b_k\}$ converges.

Step 2. Explanation.

the product of any two divergent sequence is again a divergent sequence.

So there is no such divergent sequences $\left\{a_k\right\},\left\{b_k\right\}$ such that their product $\{a_k.b_k\}$ is a convergent sequence.