Q. 10 - Calculus

Question

Give examples of sequences satisfying the given conditions or explain why such an example cannot exist.

Two convergent sequences $\left\{a_k\right\},\left\{b_k\right\}$ such that the sequence $\left\{\frac{a_k}{b_k}\right\}$ converges.

Step-by-Step Solution

Answer

Examples satisfying the given conditions is $$\begin{gathered} \left\{a_k\right\}=\left\{\frac{1}{k}\right\}, \\ \left\{b_k\right\}=\left\{1\right\} \end{gathered}$$.

Step 1. Given information.

Consider the given question,

Two convergent sequences are $\left\{a_k\right\},\left\{b_k\right\}$.

Step 2. Find if the sequences converges.

Consider the sequence $\left\{a_k\right\}=\left\{\frac{1}{k}\right\}$.

The sequence is strictly decreasing and is bounded.

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

Consider the sequence $\left\{b_k\right\}=\left\{1\right\}$.

The sequence is a constant sequence with each term equal to $1$ and is bounded.

Therefore, the sequence is convergent and converges to $1$.

Step 3. Find if the sequence satisfies the conditions.

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

The monotonically decreasing sequence which is bounded below is convergent.

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

Therefore, the sequence $\left\{\frac{a_k}{b_k}\right\}=\left\{\frac{1}{k}\right\}$ is convergent and converges to $0$.

Hence, the two convergent sequences $\left\{a_k\right\},\left\{b_k\right\}$ such that $\left\{\frac{a_k}{b_k}\right\}$ is convergent are $$\begin{gathered} \left\{a_k\right\}=\left\{\frac{1}{k}\right\}, \\ \left\{b_k\right\}=\left\{1\right\} \end{gathered}$$.