Recall from class this week:

The Integral Test:

Suppose $f$ is a coetinuous, positive, decreasing fusction on $[1,)$ where $a_n=f(n)$ for each $n1.$ Then the series ${\displaystyle \underset{n=1}{\overset{}{}}}{a}_n$ is convergent if and only if the improper integral ${\displaystyle \underset{1}{\overset{}{}}}f(x)dx$ is convergent.

$($a$)$ State the hypotheses $($or conditions$)$ for the Integral Test.

$($b$)$ State the conclusion for the Integral Test.

$($c$)$ Give a brlef explanation for why each hypothesis $($or condition$)$ is secessary for the Integral Test.

$($d$)$ Explain what the phrase $"$if and only if$"$ means in the linegral Test.