Infinite sums can be studied using improper integrals.

**harmonic sequence**, or the sequence where : If we plot the harmonic sequence, it looks like this.

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.

**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

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:

- a divergent series will remain divergent with the addition or subtraction of any finite number of terms.
- a convergent series will remain convergent with the addition or subtraction
of any finite number of terms. (Of course, the
*sum*will likely change.)

**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

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

Together, these observations give us the *integral test*:

**both converge, or both diverge**. It is impossible for one of them to diverge, and the other to converge.

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.

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!

and this only converges when .

### 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 .

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 .