Determine whether the sequence $\left\{ 2 ^ { 1 / k } \right\}$ is monotonic or eventually monotonic and whether the sequence is bounded. If the sequence is convergent, find the limit.
A) Eventually monotonic, bounded, converges to 0
B) Monotonic, unbounded, diverges
C) Monotonic, bounded, converges to 0
D) Monotonic, bounded, converges to 1

Question

Quizplus · Accepted Answer

The sequence $\left\{ 2 ^ { 1 / k } \right\}$ is monotonic because as $k$ increases, $2^{1/k}$ decreases, approaching 1. It is bounded below by 1 and above by 2 (since $2^{1/1} = 2$ and the sequence decreases from there). The limit of the sequence as $k$ approaches infinity is 1, because $2^{1/k}$ approaches 1 as $k$ becomes very large.

Determine Whether the Sequence {21/k}\left\{ 2 ^ { 1 / k } \right\}{21/k}

Determine Whether the Sequence $\left\{ 2 ^ { 1 / k } \right\}$