Q. 20 - Calculus

Question

State the converse of Theorem 7.19. Explain why Theorem 7.20 is a partial converse.

Step-by-Step Solution

Answer

The converse of the theorem is that if the sequence $\left\{a_k\right\}$ is convergent then the sequence is bounded and monotonic.

But the converse of the theorem is not true.

Step 1. Given information

We have to state the converse of Theorem 7.19.

We also have to give reason for why Theorem 7.20 is a partial converse.

Step 2. State the converse of Theorem 7.19, also give reason for why Theorem 7.20 is a partial converse.

The converse of the theorem is that if the sequence $\left\{a_k\right\}$ is convergent then the sequence is bounded and monotonic.

But the converse of the theorem is not true.

Consider the sequence $\left\{a_k\right\}=\left\{{\left(\frac{-1}{3}\right)}^k\right\}$

The sequence converges to $0$ but it is not monotonic because the sequence alternates between positive and negative-valued terms.

Theorem 7.20 is partial converse because it talks about only boundedness and not about monotonicity.