Determine whether the sequence $\left\{ \cos \left( \frac { k \pi } { 2 } \right) \right\}$ is monotonic or eventually monotonic and whether the sequence is bounded. If the sequence is convergent, find the limit.
A) Monotonic, bounded, converges to 0
B) Monotonic, not bounded, diverges
C) Monotonic, bounded, converges to 1
D) Not monotonic, bounded, diverges

Question

Quizplus · Accepted Answer

The sequence $\left\{ \cos \left( \frac { k \pi } { 2 } \right) \right\}$ is not monotonic because its values alternate between 1, 0, -1, and 0 for $k=0, 1, 2, 3,$ respectively, and then this pattern repeats. It is bounded because all its values are within the interval $[-1, 1]$. The sequence does not converge because it does not approach a single limit; instead, it keeps oscillating between 1, 0, and -1.

Determine Whether the Sequence {cos⁡(kπ2)}\left\{ \cos \left( \frac { k \pi } { 2 } \right) \right\}{cos(2kπ​)}

Determine Whether the Sequence $\left\{ \cos \left( \frac { k \pi } { 2 } \right) \right\}$