Have you ever wondered why you need integration? Integration isn't just used for finding areas or applications like particles of motion. Integration can also determine if a series converges or diverges, as in the Integral Test. But it all comes back to thinking about integrals and sums as areas, as you will see here.
The Integral Test for Series
The idea of the Integral Test is to find an integral that you can evaluate and that is a good comparison for your series. That integral allows you to figure out if the series converges or not. That looks like it might be complicated, so let's look at an example. Consider the harmonic series
.
You can think of the sum as adding up the area of rectangles, each one with width 1 and height 
.
Rectangles representing the terms of the harmonic series | StudySmarter Original
You want to find a function that hits rectangle heights from the harmonic series and that you can integrate. A natural choice is the function
.
The graph below shows the function 
and the rectangles from the harmonic series together.
Comparing the area under a function to the area of a series | StudySmarter Original
You can see that the area under the curve of 
is actually going to underestimate the area of the rectangles representing the harmonic series. Taking a look at the area under the curve,
But this is an underestimate for the sum of the harmonic series
,
which means the harmonic series must diverge.
Integral Test Requirements
In the previous example, you saw how to use an integral to show that a series diverges. You can also use an integral to show that a series converges. To do that, you would need a function that overestimates the series and also converges, which would force the series to converge as well.
There are of course some important conditions that are being used that you might not have noticed:
the function 
must be continuous (so you can integrate it);
the function 
must be positive (so you don't get cancellation of positive and negative areas that would give a false impression of the series); and
the function must be decreasing (so you have a hope of the integral having a real number as the value).
The Integral Test for Convergence and Divergence
With the previous conditions in mind, you can state the Integral Test.
Integral Test: Suppose that 
is continuous, positive, and decreasing on 
, and that 
for all 
. Then the following hold:
If
is convergent, so is
.
If
is divergent so is
.
Notice that the test is a bit more general than the discussion since you don't need to worry about underestimates or overestimates. If you look at a proof of the Integral Test you will see that fact comes from defining 
so you don't need to explicitly state it.
Proof of the Integral Test
The proof of the Integral Test follows the same procedure outlined in the example with the harmonic series, just with a general series instead of that specific series. It needs to consider that your series might start at something other than 
, and it can be quite long. Most calculus books will have the proof written out for you.
Rather than look at the proof of the Integral Test, let's look at examples of how it is used and what it can and can't do.
Integral Test Examples
First, let's look at an example of a geometric series. It is important to note that while the Integral Test can tell you if a series converges, it can't tell you what the series converges to!
For a reminder of when geometric series converge or diverge, and what they can converge to, see Geometric Series
If possible, use the Integral Test to decide if the series
converges or diverges.
Answer:
A natural function to compare the series to is
.
First, check the conditions to apply the Integral Test:
1. Is the function continuous? Yes, 
is continuous on 
.
2. Is the function positive? Yes, 
on 
.
3. Is the function decreasing? Yes, you can look at a graph to see that 
is decreasing on 
.
That means you can apply the Integral Test. Taking the integral,
So by the Integral Test, since the integral converges so does the series
.
Now, does the integral tell you what the series converges to? Not at all! For a geometric series, you know that
which is definitely not the same thing as the integral you evaluated above.
If possible, use the Integral Test to decide if the series
converges or diverges.
Answer:
The function you would choose for the Integral Test would be
.
First check that the function satisfies the conditions of the Integral Test:
1. Is the function continuous? Yes, is continuous on .
2. Is the function positive? Yes, on .
3. Is the function decreasing? Yes, but you need to be a little careful. If you look at the graph of the function it isn't always decreasing. You can use the First Derivative Test to show that is decreasing on 
. Since the function is eventually decreasing that is all that you need to be able to apply the Integral Test.
Since all of the conditions are met to apply the Integral Test, taking the integral gives
To evaluate that integral, use the substitution 
, so 
, or
.
Then
Substituting that back in,
So by the Integral Test,
converges.
This does NOT tell you that the series converges to 1/2. It only tells you that it converges. Remember from the areas in the example at the start of this explanation, the area of the integral and the area of the series are not going to be the same.
For a review on using the derivative to see if a function is increasing or decreasing, see The First Derivative Test. To learn more about integration techniques such as in the example above, see Integration by Substitution.
If possible, use the Integral Test to decide if the series
converges or diverges.
Answer:
The problem in using the Integral Test on this series is that it isn't always positive. It is an alternating series. So what happens if you try and take
?
First, it isn't always positive. So right away you know you can't use the Integral Test. You don't even need to check and see if it is decreasing or continuous. That means the Integral Test can't be applied to this series.
If possible, use the Integral Test to see if the series
converges or diverges.
Answer:
Notice that this is a P-series with 
, so you already know it diverges. But it is good to check and see that the Integral Test gives you the correct result as well. To do that choose the function
.
Checking the three conditions to apply the Integral Test:
1. Is the function continuous? Yes, is continuous on 
.
2. Is the function positive? Yes, on 
.
3. Is the function decreasing? Yes, you can look at a graph to see that is decreasing on 
.
Since all of the conditions are met you are ready to take the integral:
Since the integral diverges, by the Integral Test so does the series
.
Integral Test - Key takeaways
- The Integral Test compares a series to an integral in order to see if a series converges or diverges.
- The three conditions that the function you are integrating needs to meet are:
the function must be continuous
the function must be positive
the function must be decreasing
Integral Test: Suppose that is continuous, positive, and decreasing on , and that for all . Then the following hold:
If
is convergent, so is
.
If
is divergent so is
.
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