Using the integral test, determine whether the infinite series
converges or diverges, or if the test cannot be applied.
First we must establish whether or not the test can be applied. If we let
then we much check if f(x) is continuous, decreasing, and positive.
Step (1): Check Continuity
Both the numerator and denominator are continuous functions on (2,∞), so their ratio will also be continuous. So f is continuous.
Step (2): Check Positivity
Both the numerator and denominator are positive functions on (2,∞), so their ratio will also be positive. So f is positive.
Step (3): Check to See if f is Decreasing
We can apply the usual first derivative test:
The above quantity is negative for
so we have that our function is also decreasing on this interval.
Step (4): Apply Integral Test
Since f is positive, continuous, and decreasing, we can apply the integral test.
Therefore, the infinite series diverges, because the above integral diverges.
Discussion of Each Step
Step (1) and (2)
These checks must be done, but are, in this example, straightforward.
Recall that the first derivative test tells us that a function is decreasing on an interval if the first derivative of that function is negative everywhere on that interval. Using the quotient rule to calculate our derivative, we find that indeed, the function is decreasing on (2,∞).
Given that the question asks us to apply the integral test, we should know immediately how to get started: check for continuity, positivity, and if our function is decreasing. These three checks must always be performed before the test can be applied.