Tests of Convergence

It is very easy to see that a simple improper integral may be very hard to decide whether it is convergent or divergent. For example, the improper integral

displaymath231

is hard to study since it is very difficult to find an antiderivative of the function tex2html_wrap_inline233 . The tests of convergence are very useful tools in handling such improper integrals. Unfortunately some improper integrals fails to fall under the scope of these tests but we will not deal with them here.

Recall the p-Test: Regardless of the value of the number p, the improper integral

displaymath237

is always divergent. Moreover, we have

tex2html_wrap_inline239
tex2html_wrap_inline241 is convergent if and only if p <1
tex2html_wrap_inline239
tex2html_wrap_inline247 is convergent if and only if p >1

Note that one may generalize this test to include the following improper integrals

displaymath251

The conclusion is similar to the above one. Indeed we have

tex2html_wrap_inline239
tex2html_wrap_inline255 is convergent if and only if p <1
tex2html_wrap_inline239
tex2html_wrap_inline261 is convergent if and only if p <1

Comparison Test Let f(x) and g(x) be two functions defined on [a,b] such that

displaymath271

for any tex2html_wrap_inline273 . Then we have

tex2html_wrap_inline239
If tex2html_wrap_inline277 is convergent, then tex2html_wrap_inline279 is convergent.
tex2html_wrap_inline239
If tex2html_wrap_inline279 is divergent, then tex2html_wrap_inline277 is divergent.

Example. Decide on the convergence or divergence of

displaymath231

Answer. We have for tex2html_wrap_inline289

displaymath291

The p-Test implies that the improper integral tex2html_wrap_inline293 is convergent. Hence the Comparison test implies that the improper integral

displaymath231

is convergent.

We should appreciate the beauty of these tests. Without them it would have been almost impossible to decide on the convergence of this integral.

Before we get into the limit test, we need to recall the following:
we will say and write tex2html_wrap_inline297 when tex2html_wrap_inline299 if and only if

displaymath301

Limit test Let f(x) and g(x) be two positive functions defined on [a,b]. Assume that both functions exhibit an improper behavior at a and tex2html_wrap_inline297 when tex2html_wrap_inline313 , then we have
tex2html_wrap_inline315 is convergent if and only if tex2html_wrap_inline317 is convergent.

This statement is still valid whether a is a finite number or infinite or if the improper behavior is at b.

Example. Establish the convergence or divergence of

displaymath323

Answer. Clearly this integral is improper since the domain is unbounded (Type II). Moreover since the function tex2html_wrap_inline325 is unbounded at 0, then we also have an improper behavior at 0. First we must split the integral and write

displaymath327

First let us take care of the integral tex2html_wrap_inline329 . Since

displaymath331

when tex2html_wrap_inline333 , and (because of the p-test) the integral

displaymath335

is convergent, we deduce from the limit test that

displaymath337

is convergent. Next we investigate the integral tex2html_wrap_inline339 . Since

displaymath341

when tex2html_wrap_inline343 , and (because of the p-test) the integral

displaymath345

is convergent, we deduce from the limit test that

displaymath347

is convergent. Therefore, the improper integral

displaymath323

is convergent.

Remark. One may notice that in the above example, we only used the limit test combined with the p-test. But we should keep in mind that it is not the case in general. The next example shows how the use of other tests is more than useful.

Example. Establish the convergence or divergence of

displaymath351

Answer. Again it is easy to see that we have an improper behavior at both 0 and tex2html_wrap_inline353 . Hence we must split the integral and write

displaymath355

The integral tex2html_wrap_inline357 is easy to take care of since we have

displaymath359

and because tex2html_wrap_inline361 is convergent (by the p-test), the basic comparison test implies that

displaymath363

is convergent. Next we take care of the integral tex2html_wrap_inline365 . Here we use the limit test. Indeed, since tex2html_wrap_inline367 when tex2html_wrap_inline333 , then we have

displaymath371

Because tex2html_wrap_inline373 is divergent (by the p-test), then the limit test implies that the integral

displaymath375

is divergent. Conclusion the improper integral

displaymath351

is divergent.

Remark. One may argue that the above example is in fact not a good one to illustrate the use of different tests. Since if we have showed first that the integral

displaymath375

is divergent via the limit test, then we do not need to take care of the other integral and conclude to the divergence of the given integral. A very good point. Now consider the improper integral

displaymath381

and show that in this case the integral is convergent. Let us point out that the trigonometric functions are very bad when it comes to look at what is happening at tex2html_wrap_inline383 . Hence the limit test is absolutely not appropriate to use...

Example. Establish the convergence or divergence of

displaymath385

Answer. This is clearly not an improper integral of Type II. Let us check if it is of Type I. First notice that tex2html_wrap_inline387 . Hence the function is unbounded at x=1 and x=3 (you must check it by taking the limit.. left as an exercise). Since 3 is between 2 and 4, we deduce that the integral is improper and the only bad point is 3. Hence we must split the integral to get

displaymath395

Let us take care of the integral . It is easy to see that when tex2html_wrap_inline399 , then we have

displaymath401

The p-test implies that the integral

displaymath403

is convergent. Hence by the limit test we conclude that the integral

displaymath405

is convergent. Using the same arguments, we can show that the integral

displaymath407

is also convergent. Therefore the integral

displaymath385

is convergent.

Note that all the tests so far are valid only for positive functions. One may then wonder what happens to improper integrals involving non positive functions. A partial answer is given by the Absolute Convergence tests.

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Mohamed A. Khamsi
Tue Dec 3 17:39:00 MST 1996

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