Q. 4 - 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 $\left\{a_k\right\}$ converges.

Step-by-Step Solution

Answer

The two convergent sequences are $\left\{a_k\right\}=\left\{1\right\} and \left\{b_k\right\}=1$.

Step 1. Given Information.

The sequences are convergent.

Step 2. Consider the sequence.

Consider the sequence,

$\left\{a_k\right\}=\left\{1\right\}; \left\{b_k\right\}=1$

both the sequences are constant and bounded and converges to one.

Step 3. Addition of the sequence.

The sequence,

${\{a_k+b\}}_k=\left\{2\right\}$

which is again constant and bounded and converges to two.