Absolute Convergence of Improper Integrals

Since most of the tests of convergence for improper integrals are only valid for positive functions, it is legitimate to wonder what happens to improper integrals involving non positive functions. First notice that there is a very natural way of generating a positive number from a given number: just take the absolute value of the number. So consider a function f(x) (not necessarily positive) defined on [a,b]. Then let us consider the positive function |f(x)| still defined on [a,b]. It is easy to see that both functions f(x) and |f(x)| will exhibit the same kind of improper behavior. Therefore, one may ask naturally what conclusion do we have if we know something about the integral

displaymath180

We have the following partial answer:

If the integral tex2html_wrap_inline182 is convergent, then the integral tex2html_wrap_inline184 is also convergent.

We have to be careful the converse is not true. Indeed, the improper integral

displaymath186

is convergent while the improper integral

displaymath188

is divergent. This is quite hard to show. On the other hand, it shows that the convergence of tex2html_wrap_inline190 carries more information than just convergence. In this case, we say that the improper integral tex2html_wrap_inline192 is absolutely convergent. And if the improper integral tex2html_wrap_inline192 is convergent while the improper integral tex2html_wrap_inline190 is divergent, we say it is conditionally convergent.

Example. Establish the convergence or divergence of

displaymath198

Answer. We have an improper integral of Type II. Since the function tex2html_wrap_inline200 is not positive on tex2html_wrap_inline202 , we will investigate whether the given improper integral is absolutely convergent. Hence we must consider the improper integral

displaymath204

Let us check whether we have a Type I behavior. Clearly the point 0 is a bad point. We leave it as an exercise to check that the function tex2html_wrap_inline206 is indeed unbounded around 0. So we must split the integral and write

displaymath208

First let us take care of the integral

displaymath210

We know that tex2html_wrap_inline212 when tex2html_wrap_inline214 . Hence we have

displaymath216

when tex2html_wrap_inline218 . Since the integral tex2html_wrap_inline220 is convergent via the p-test, the limit test enables us to conclude that the integral

displaymath210

is convergent. Next we take care of the improper integral

displaymath224

We can not use the limit test since the function tex2html_wrap_inline226 does not have a nice behavior around tex2html_wrap_inline228 . But we know that tex2html_wrap_inline230 for any number x. Hence we have

displaymath234

for any tex2html_wrap_inline236 . Since the improper integral tex2html_wrap_inline238 is convergent via the p-test, the basic comparison test implies that the improper integral

displaymath224

is convergent. Therefore putting the two integrals together, we conclude that the improper integral

displaymath204

is convergent. This clearly implies that the improper integral

displaymath198

is absolutely convergent.

Example. Show that the improper integral

displaymath186

is convergent.
Answer. As we mentioned before, this improper integral is not absolutely convergent. So there is no need of considering the absolute value of the function. Note that the integral is improper obviously because of tex2html_wrap_inline228 . 0 is not a bad point since

displaymath250

But even if it is not a bad point, we will isolate it by writing

displaymath252

The integral tex2html_wrap_inline254 is not improper. So we concentrate on the integral

displaymath256

We know by definition that

displaymath258

Now consider the proper integral tex2html_wrap_inline260 . An integration by parts gives

displaymath262

Since

displaymath264

and

displaymath266

we get

displaymath268

Note now that the improper integral tex2html_wrap_inline270 is in fact absolutely convergent. Indeed, we have

displaymath272

and since by the p-test the improper integral tex2html_wrap_inline274 is convergent, the basic comparison test implies the desired conclusion, that is tex2html_wrap_inline276 is convergent. Therefore the improper integral tex2html_wrap_inline270 is convergent. Since

displaymath268

then the improper integral tex2html_wrap_inline282 is convergent.

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

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