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Wednesday, 27 March 2013

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
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