Q. 25 - Calculus

Question

For each of the sequences in Exercises 23–52 determine whether the sequence is monotonic or eventually monotonic and whether the sequence is bounded above and/or below. If the sequence converges, give the limit.

25. $\left\{\sin \left(\frac{k\mathrm{\pi }}{2}\right)\right\}$

Step-by-Step Solution

Answer

The sequence is not monotonic and bounded and not convergent.

The sequence has no limit.

Step 1. Given information

We have been given the sequence $\left\{\sin \left(\frac{k\mathrm{\pi }}{2}\right)\right\}$.

Step 2. Determine whether the sequence is monotonic or eventually monotonic and whether the sequence is bounded above and/or below.

The sequence $\left\{a_k\right\}=\left\{\sin \left(\frac{k\mathrm{\pi }}{2}\right)\right\}$ is not a monotonic sequence because sign of $\sin \left(\frac{k\mathrm{\pi }}{2}\right)$ varies as $k$ increases.

The sequence $\left\{a_k\right\}=\left\{\sin \left(\frac{k\mathrm{\pi }}{2}\right)\right\}$ is a bounded sequence because

$-1\le \sin \left(\frac{k\mathrm{\pi }}{2}\right)\le 1$

The given sequence has upper and lower bound, therefore, the sequence is bounded.

The sequence is not monotonic but is bounded. Therefore, the sequence is not convergent and has no limit.