Infinite sums can be studied using improper integrals.

We have seen that we can graph a sequence as a collection of points in the plane. For instance, consider the harmonic sequence, or the sequence where : If we plot the harmonic sequence, it looks like this.
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Is there a nice way to visualize the sum One way to visualize the sum is to make rectangles whose areas are equal to the terms of the sequence.

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The entire area above is exactly equal to the sum: The previous image should remind you of a Riemann Sum, and for good reason. This technique lets us visually compare the sum of an infinite series to the value of an improper integral, as long as we imagine that we can draw infinitely many such rectangles. For instance, if we add a plot of to our picture above
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we see that This leads us to an interesting observation.

Let be continuous, positive, and decreasing on with . If diverges, so does .

That’s a pretty good observation, but we can do better. Consider the sequence : Think about it. If diverges, then so does Why? The second sum is simply the first sum missing and no single number can be responsible for this sum diverging to infinity. This fact can be summed up in the following theorem:

Back to the task at hand! Let’s graph using our rectangle technique.

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The shaded area above is exactly equal to the sum We again visually compare the sum of an infinite series to the value of an improper integral. For instance, if we add a plot of to our picture above
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we see that This conclusion leads us to another interesting observation.

Let be continuous, positive, and decreasing on with . If diverges, so does .

Together, these observations give us the integral test:

Notice we’ve also included the observation that a finite number of terms cannot affect the convergence or divergence of a series. In particular, our function can be different from our sequence for a finite number of terms.

Does the harmonic series converge or diverge?
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By the integral test, converges if and only if converges

Thus the Harmonic series must diverge.

Our last example is a ‘‘theorem’’ in some textbooks. We think it is just an application of the integral test. However, you’ll want to be able to apply this result to other examples!

Estimating Series Using Improper Integrals

Another cool application of this graphical viewpoint on series involves estimating the sum of a series.

We cannot yet compute exactly, and we will not learn how to do so in this course.

Say we wanted to approximate this sum within an error of . How many terms should we sum? Is it enough to sum the first ten terms? The first hundred? We want to find a number where we can be sure that Again, we’re looking for a number so that the difference between the actual (but unknown) sum of the series and the -th partial sum is less than . Now, Set to be This is often called the ‘‘remainder’’ or ‘‘tail’’ of the series, because it is what is left over after we remove the -th partial sum. Consider the following graph, where the rectangles correspond to the terms of .

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Shifting these rectangles over one unit, we have the following.
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Now we know In other words, the remainder of the series must be less than the given integral. So, if we can find a whole number for which we will be sure that we have summed enough terms in the series to get to within .

This shows that if we sum terms of the series , we will get within of the true answer.

Using a computer to sum the first terms we get Using advanced concepts like Fourier Analysis, it turns out this series actually converges to , and we know that . The difference between our calculated value and the actual value is just barely less than .