Problem 1 is Exercise 7.7.64 from the book. It asks to determine the convergence of

There are several ways of approaching this problem.

Method 1: Begin by noting that

and so

can be evaluated by using the substitution so which transforms the integral into

that can be solved with the trig. substitution so and

Now we return to the problem:

We have found the value of the integral, so in particular, it converges.

Method 2: First,

We use comparison for both and

If Since

we have by comparison that is finite.

If Since

we have by comparison that is also finite.

Hence, converges.

Method 3: As before,

The change of variables transforms into

and

which of course is the same as

So all we need to do is to show that converges, and for this we can use comparison, because

and

Problem 2 is an easier version of Exercise 7.7.63 from the book. It asks to determine the convergence of

The problem here is that we cannot find so we are forced to use comparison. A natural thing to try would be to compare with

Since we have

The problem is that has an asymptote at 0. However, we can split the integral as

The first integral is finite, simply because it is the integral of a continuous function on some finite interval, there is nothing improper here. The second converges by comparison with the -integral with

It follows that the whole integral converges.

A slightly different approach is to use limit comparison rather than direct comparison:

It follows that converges because does; and we have to treat

as above. We cannot just compare with (which happens to diverge), because the limit comparison test requires that the functions that we compare are continuous, and is not continuous at 0.

Typeset using LaTeX2WP. Here is a printable version of this post.

43.614000-116.202000

Advertisements

Like this:

LikeLoading...

Related

This entry was posted on Friday, November 13th, 2009 at 2:32 pm and is filed under 175: Calculus II. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

The only reference I know for precisely these matters is the handbook chapter MR2768702. Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. In Handbook of set theory. Vols. 1, 2, 3, 1951–2119, Springer, Dordrecht, 2010. (Particularly, section 7.) For closely related topics, see also the work of Yong Cheng (and of Cheng and Schindler) on Harr […]

As other answers point out, yes, one needs choice. The popular/natural examples of models of ZF+DC where all sets of reals are measurable are models of determinacy, and Solovay's model. They are related in deep ways, actually, through large cardinals. (Under enough large cardinals, $L({\mathbb R})$ of $V$ is a model of determinacy and (something stronge […]

Throughout the question, we only consider primes of the form $3k+1$. A reference for cubic reciprocity is Ireland & Rosen's A Classical Introduction to Modern Number Theory. How can I count the relative density of those $p$ (of the form $3k+1$) such that the equation $2=3x^3$ has no solutions modulo $p$? Really, even pointers on how to say anything […]

(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain […]

It is not possible to provide an explicit expression for a non-linear solution. The reason is that (it is a folklore result that) an additive $f:{\mathbb R}\to{\mathbb R}$ is linear iff it is measurable. (This result can be found in a variety of places, it is a standard exercise in measure theory books. As of this writing, there is a short proof here (Intern […]

Let $s$ be the supremum of the $\mu$-measures of members of $\mathcal G$. By definition of supremum, for each $n$, there is $G_n\in\mathcal G$ with $\mu(G_n)>s-1/n$. Letting $G=\bigcup_n G_n$, then $G\in \mathcal G$ since $\mathcal G$ is closed under countable unions, and $\mu(G)=s$, since it is at least $\sup_n\mu(G_n)$ but it is at most $s$ (by definiti […]

The result you are trying to prove is false. For example, if $a=\omega+1$ and $b=\omega+\omega$, then $a+b=\omega\cdot 3>b$. Here is what is true: first, the key result you should establish (by induction) is that An ordinal $\alpha>0$ has the property that for all $\beta

Very briefly: Yes, there are several programs being developed that can be understood as pursuing new axioms for set theory. For the question itself of whether pursuing new axioms is a reasonably line of inquiry, see the following (in particular, the paper by John Steel): MR1814122 (2002a:03007). Feferman, Solomon; Friedman, Harvey M.; Maddy, Penelope; Steel, […]

This is a very interesting question and the subject of current research in set theory. There are, however, some caveats. Say that a set of reals is $\aleph_1$-dense if and only if it meets each interval in exactly $\aleph_1$-many points. It is easy to see that such sets exist, have size $\aleph_1$, and in fact, if $A$ is $\aleph_1$-dense, then between any tw […]