Example: Integral Test with a Logarithm

Problem

Using the integral test, determine whether the infinite series

converges or diverges, or if the test cannot be applied.

Complete Solution

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.

Step (3)

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,∞).