Q. 5 - Calculus

Question

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

Step-by-Step Solution

Answer

There is no such sequence $\left\{a_k\right\},\left\{b_k\right\}$ such that $\{a_k-b_k\}$ is divergent.

Step 1. Given Information.

Two convergent sequences $\left\{a_k\right\}$ and $\left\{b_k\right\}$ such that the sequence $\{a_k-b_k\}$ diverges.

Step 2. Explanation.

If two sequences are convergent, the sum and difference of the sequence is always convergent.

There is no such sequence $\left\{a_k\right\},\left\{b_k\right\}$ such that $\{a_k-b_k\}$ is divergent.