Q. 7 - 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 $\{a_k.b_k\}$ diverges.

Step-by-Step Solution

Answer

The examples of such divergent sequence is $\left\{a_k\right\}=\left\{k\right\}$ and $\left\{b_k\right\}=\left\{k\right\}$.

Step 1. Given Information.

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

Step 2. Consider the sequence.

Consider the sequence,

$$\begin{gathered} \left\{a_k\right\}=\left\{k\right\} \\ \left\{b_k\right\}=\left\{k\right\} \end{gathered}$$

Both are monotonically increasing function and are not bounded above.

So the sequences are divergent.

Step 3. Explanation of the statement.

The product of the sequence $\{a_k.b_k\}=\left\{k^2\right\}$ is again a strictly increasing sequence and not bounded above.

So the resultant sequence is also divergent.