Recent content by Gib Z

Logged back in after a long period of inactivity solely to respond to WannabeNewton's review of this book. This is a great book for people who have just learned multivariable calculus, which I believe is the intended audience. mathwonk has already outlined its positive attributes. Spivak demands...

Hint 1: \int_{\gamma} f(z) dz = \int^b_a f( \gamma(t) ) \gamma'(t) dt. Hint 2: \cos(\sin x - x) = \operatorname{Re} e^{i\sin x - ix}.

I may have, but I have a better question now. If a_n is a monotone decreasing sequence such that \sum a_n converges, show na_n \to 0, and then make the generalization that a measure theorist would make.

A good place to start for a true beginner is Apostol's "Mathematical Analysis". If you feel comfortable with much of that content or are up for a challenge, you could begin straight away with Royden's Real analysis textbook. It covers all the basics (sequences, series, uniform convergence and...

There isn't really a protypical situation to use u = \cos x + i \sin x like there are for other substitution, eg trying t=\tan x if one sees \sqrt{1+t^2} in the integrand. Thing will become for clear when you learn that this mysterious function cis(x) is actually just e^{ix} and all the...

http://www.wolframalpha.com/input/?i=sum+k%3D1+to+k%3D100+1%2Fk

Pretty close, hopefully the mistake was just a typo: "Then, however, v< 1-(-1)^n /n for any n in N." Replace "any" with "some" and it's fine. Can you see why? EDIT: Too slow lol.

Draw a new diagram. Also keep in mind the fact arg(p/q) = arg(p) - arg(q) . You should be able to see from the diagram why arg(z-a) - arg(z-b) is the angle <azb (on one side of the chord connecting ab at least). It then follows from the quoted theorem that the locus is the arc of a semi-circle.

The useful result from Euclidean Geometry is the following: Angles at the circumference standing on equal chords are equal.

In another thread last night you were studying the analogous result for equivalent metrics. This follows from that result, as the norm induces a metric.

Yes, it is.

Susanne217- This would be categorized as the theory of Integral equations. Here, specifically, the theory borrows tools from the theory of Metric spaces, in particular the Banach Contraction Mapping theorem. OP - I don't know how to use Maple, so I'm not sure how this question was intended to...

In the case that it is what I thought, then it's not as simple as recognizing it as a pre-prepared Riemann sum. With some careful estimates to bound the sum, you should get the result to be 1/3.

Please take more care to expressing to others (and yourself) what it is you want to find. As written the sum doesn't make total sense. It could be what Susanne217 said above, or Riemann sum comment makes me think you could have also meant \displaystyle\lim_{n\to\infty} \sum_{k=1}^n...

The definition of equivalent metric actually doesn't imply that inequality, you have the implication reversed. That inequality is a sufficient, but not necessary condition for two metrics to be equivalent. Two exercises you can try are to prove that it is a sufficient condition, and to verify...