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Convergence Tests 1

Unlike the nth term test for divergence from the last module, this module gives several tests which, if successfully applied, give a definitive answer of whether a series converges or not. A common feature of the tests in this module is that they use a comparison.

Integral test for convergence and divergence

This is a test which can definitively tell whether a series converges or diverges. However, it can be harder to apply.

Integral test for convergence and divergence

If is a positive, decreasing function, then for any integer ,

The double arrow means if and only if. So the series and integral either both converge, or they both diverge.

To see why, visualize the series as a sum of rectangles with base 1 and height . If these rectangles are drawn to the right of the curve , then the result is the following figure. Each rectangle is labeled with its area. The combined area of the rectangles completely contains the area under , which establishes the inequality shown:

On the other hand, if the rectangles are drawn to the left of the curve , then all the rectangles lie below the curve. Their combined area is less than the area under the curve , which establishes the inequality shown:

Combining these two inequalities gives

If the integral diverges, then the series diverges (by the first inequality). And if the integral converges, then the series converges (by the second inequality). This establishes the integral test.

Example

Use the integral test to determine if

converges or diverges.

Computing, one finds (using the u-substitution ) that

which diverges since as . Since the integral diverges, the series also diverges by the integral test.

Example

Use the integral test to determine if

converges or diverges.

The integral test says the series converges if and only if

converges. We know this integral converges (recall that is the PDF for the exponential distribution). But we can also compute it again.

This integral is a good candidate for integration by parts, with

Thus,

Since the integral converges, the series converges too, by the integral test.

The p-series test

The next example is important enough that it gets its own name: the p-series. This makes use of the p-integrals that we computed earlier.

Example

Find the values of for which the series

converges.

We know from the integral test that

converges if and only if

converges. But this integral converges if and only if , as we saw in the module on p-integrals. Therefore, the p-series converges if and only if .

The p-series test

The p-series converges if and only if .

Harmonic series

The series , known as the harmonic series, diverges by to the p-series test. This is a significant fact to keep in mind because the harmonic series diverges even though the terms of the series go to 0. Thus, the harmonic series is a demonstration that the nth term test is a test for divergence only and cannot be used to show a series converges.

Note that the harmonic series is a sort of boundary between convergence and divergence. The series diverges, but the series converges.

Example

The p-series test proves the convergence of two examples from the last module: and .

Example

Determine the values of for which the series

converges.

Making the substitution

we see that

This integral (again from our knowledge of the p-integral) converges if and only if . Therefore, by the integral test, the series

converges if and only if .

Comparison test

The integral test compared a series to its related integral. This test compares one series to another.

Comparison test

Let and be positive sequences such that for all . It follows that

  1. if converges, then converges.
  2. if diverges, then diverges.

In other words, if a series is smaller than a convergent series, then it converges too. If a series is bigger than a divergent series, then it diverges too.

Caveat

It is critical that the inequality be in the correct direction. A series which is larger than a convergent series might converge or diverge. A series which is smaller than a divergent series might converge or diverge.

Example

Show that diverges.

Note that . Since diverges, the series diverges too, by the comparison test.

Example

Show that

converges.

Note that

since the numerator on the left is smaller, and the denominator on the left is bigger. So

converges (p-series test from above), and so

converges as well, by the comparison test.

Example

Determine whether

converges or diverges. Hint: try for a rough upper bound or lower bound on and see which one gives the right comparison.

A lower bound for might be (or any exponential). This gives

which diverges, since it is a constant multiple of the harmonic series. This comparison does not go in the right direction, since our original series is smaller than a divergent series. Thus, we should try going in the other direction to find an upper bound for .

A rough upper bound for is . This gives

This series diverges (see the example earlier in this module). Therefore, the original series, which is bigger than a divergent series, also diverges.

Limit test

The final test of this module is a slightly different type of comparison. Recall that when comparing two functions and to see which is "bigger" asymptotically, one computes the limit

If this limit is infinite, then is bigger. If the limit is 0, then is bigger. If the limit is where , then the two functions are roughly equal (up to a constant multiple). It is this third case that is used for this test (sometimes called the Limit comparison test):

Limit test

Let and be positive series. If and , then the series and either both converge or both diverge.

The key to the limit test is finding a suitable sequence which is approximately equal to in the limit. Often, will be a ratio, in which case the lower order terms in the numerator and denominator can be dropped, and what is left will be . Ideally, it will be easy to see if converges or diverges. Finally, one must check that .

Example

Show that

diverges.

By dropping the lower order terms in the numerator and denominator, one finds

So is a good choice. Assuming the above approximation is not too rough, the work is done since the harmonic series, , diverges.

To make sure the approximation is not too rough, compute the limit

Thus, by the limit test, and either both converge or both diverge. Since diverges, so too must .

Example

Determine whether

converges or diverges.

With an unusual series like this, the nth term test is a good first thing to try. But it is inconclusive since as .

To get a handle on how this function acts, note that when is large, is small, and so we can use the Taylor series for about 0:

Thus, , which means is a good candidate for in the limit test. This will work:

Since diverges (harmonic series), it follows by the limit test that diverges also.


EXERCISES

  • Determine whether the following series converges or diverges