Use the ratio test in Theorem 7.6 to analyze the monotonicity of the sequence $\left\{ \frac { 2 k ^ { 2 } } { k ! } \right\}$
A) Strictly increasing
B) Eventually increasing
C) Strictly decreasing
D) Eventually decreasing

Question

Quizplus · Accepted Answer

The ratio test involves taking the limit of the absolute value of the ratio of the (k+1)th term to the kth term. For the given sequence, this ratio is $\frac{2(k+1)^2}{(k+1)!} \cdot \frac{k!}{2k^2} = \frac{2(k+1)^2}{(k+1)2k^2} = \frac{k+1}{k^2}$. As k increases, this ratio decreases towards 0, indicating that the sequence is eventually decreasing.

Use the Ratio Test in Theorem 7 {2k2k!}\left\{ \frac { 2 k ^ { 2 } } { k ! } \right\}{k!2k2​}

Use the Ratio Test in Theorem 7 $\left\{ \frac { 2 k ^ { 2 } } { k ! } \right\}$