Consider the infinite series
Determine whether it is convergent using the integral test.
Step 1: Pull Out the Negative Sign
If S converges, then the given infinite series converges.
Step 2: Check to see if the integral test can be applied
Let . Then
- f(x) is continuous
- f(x) is decreasing
- f(x) is non-negative
Therefore the integral test can be applied.
Step 3: Apply the Integral Test
Step 4: Conclusion
The result is finite, so S is convergent by the integral test, so the given series is also convergent.
Explanation of Each Step
If we did not pull out the negative sign, we would not be able to apply the integral test, because this test can only be applied when all terms in the series are positive. This simple algebraic manipulation allows us to apply the integral test.
There are only three criteria we need to check before applying the integral test. Because all three criteria are met, we can apply this test.
In Step (3) we applied the formula for the integral test, using the method of integration by parts to calculate the integral.
Possible Challenge Areas
In the example we were given, we only had to pull out a negative sign, but what if we were asked to determine whether
converges? How would we approach this problem?
Observing that the function
is strictly positive for x>a, we can pull out the first terms of the sum that are negative, knowing that the remaining terms are positive, and can be used in the integral test.