Recall from class this week:

The Integral Test:

Suppose f is a continuous, positive, decreasing function on $[1,$ $)$ where an $=$ f $($n$)$ for

each n $1.$ Then the series

$$

n$=1$

an is convergent if and only if the improper integral

$$

$1$ 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 brief explanation for why each hypothesis $($or condition$)$ is necessary for the

Integral Test.

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