Q. 12 - Calculus

Question

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

Explain what is important about monotonic and bounded sequences.

Step-by-Step Solution

Answer

If the sequence is decreasing and is bounded below then the sequence is convergent and if the decreasing sequence is not bounded below then the sequence is divergent.

If the sequence is increasing and is bounded above then the sequence is convergent and if the increasing sequence is not bounded below then the sequence is divergent.

Step 1. Given information.

Consider the given question,

The sequence is monotonic and bounded.

Step 2. Explain the monotonicity and boundedness of the sequence.

Consider the monotonicity and boundedness of the sequence.

The monotonicity and boundedness of the sequence helps in determining the convergence and divergence of the sequence.

If the sequence is decreasing and is bounded below then the sequence is convergent and if the decreasing sequence is not bounded below then the sequence is divergent.

If the sequence is increasing and is bounded above then the sequence is convergent and if the increasing sequence is not bounded below then the sequence is divergent.

Thus, both the monotonicity and boundedness of the sequence is important for determining the behavior of the sequence.