Provide an appropriate response.
-Which of the following sequences do not meet the conditions of the Integral Test?
I. $a _ { n } = n ( \sin n + 1 )$
II. $\mathrm { a } _ { \mathrm { n } } = \frac { 1 } { \mathrm { n } \mathrm { P } + \mathrm { p } }$
III. $a _ { n } = \frac { 1 } { n \sqrt { n } }$
A) I only
B) I and III
C) I, II, and III
D) II and III

Question

Quizplus · Accepted Answer

Explanation: The Integral Test can be applied to a sequence $a_n$ if the function $f(n) = a_n$ is continuous, positive, and decreasing for all $n$ greater than or equal to some number $N$. For sequence I, $a_n = n(\sin n + 1)$, the function is not always positive due to the $\sin n$ term, which oscillates between -1 and 1. Therefore, it does not meet the conditions for the Integral Test. Sequence II, $a_n = \frac{1}{nP + p}$, and sequence III, $a_n = \frac{1}{n\sqrt{n}}$, both define functions that are continuous, positive, and decreasing for $n > 0$, assuming $P$ and $p$ are positive constants in sequence II. Thus, they meet the conditions for the Integral Test.

Provide an Appropriate Response an=n(sin⁡n+1)a _ { n } = n ( \sin n + 1 )an​=n(sinn+1)

Provide an Appropriate Response $a _ { n } = n ( \sin n + 1 )$