Use the derivative test in Theorem 7.6 to analyze the monotonicity of the sequence $\left\{ k ^ { 3 } - k \right\}$ Assume that the first term has index 1.
A) Strictly decreasing
B) Eventually decreasing
C) Strictly increasing
D) Eventually increasing

Question

Quizplus · Accepted Answer

The derivative of the sequence's nth term with respect to n, $f'(n) = 3n^2 - 1$, is always positive for $n > \frac{1}{\sqrt{3}}$. Since the sequence starts with an index of 1, and $1 > \frac{1}{\sqrt{3}}$, the sequence is strictly increasing from the first term onwards.

Use the Derivative Test in Theorem 7 {k3−k}\left\{ k ^ { 3 } - k \right\}{k3−k}

Use the Derivative Test in Theorem 7 $\left\{ k ^ { 3 } - k \right\}$