Use the difference test in Theorem 7.6 to analyze the monotonicity of the sequence $\left\{ \frac { k } { k + 3 } \right\}$
A) Strictly decreasing
B) Eventually decreasing
C) Eventually increasing
D) Strictly increasing

Question

Quizplus · Accepted Answer

To analyze the monotonicity of the sequence $\left\{ \frac { k } { k + 3 } \right\}$, we calculate the difference between consecutive terms: $\frac { k+1 } { k+4 } - \frac { k } { k+3 }$. Simplifying this expression gives $\frac { 3 } { (k+4)(k+3) }$, which is always positive for all $k$. This means the sequence is strictly increasing.

Use the Difference Test in Theorem 7 {kk+3}\left\{ \frac { k } { k + 3 } \right\}{k+3k​}

Use the Difference Test in Theorem 7 $\left\{ \frac { k } { k + 3 } \right\}$