Recent content by TPAINE

I don't see the problem. The slope is -3/100 at the point (4,f(4))=(4,.4).

Excellent exposition. Thank you. To be clear, in your third paragraph you are using a limsup/liminf argument?

For example, consider 1-1+1/2+1-1-1/3+1-1+1/4+1-1-1/5... This diverges, because if it had a limit L, choose epsilon to be 1/3. Then at some point, the alternating 1,-1s are going to make the sequence be more than 1/3 away from L. But if we try to naively group them, as you propose above, we get...

I'm assuming you are proposing grouping the positive terms and using the alternating series test. I'm not sure how I would rigorously justify using that. That would would show that, for positive integer n, the partial sums of the first 3n and 3n+2 terms converge, but it says nothing about the...

It's monotonic, so if you show that it is bounded, you're in business. Now by the comparison test, we have n!>n^2 for sufficiently large n, so take reciprocals and go from there.

Do what Maxter said and use contradiction. If lime g(x) existed, the limit of the sum would exist. Therefore, the limit of g(x) does not exist.

Homework Statement Prove 1+1/3-1/2+1/5+1/7-1/4+1/9+1/11-1/6... converges. 3. Relevant Info This was left as an exercise to the reader in the text of Rudin. Both the root and ratio tests fail, and I don't see anything obvious. I found a proof online that it converges to 3/2 log 2, but...

Homework Statement I am trying to solve part d of problem 9 in chapter 2 of Rudin's Principles of Mathematical Analysis. The problem is: Let E* denote the set of all interior points of a set E (in a metric space X). Prove the complement of E* is the closure of the complement of E. I will...

I'm aware it can be done that way. However, is there a way to do it without separating the curve into branches, for equations where that is not so easy (or impossible)?

Homework Statement Find the slopes of the two tangent lines of x^3-y^2+x^2=0 at 0,0. Homework Equations Differentiating implicitly we get (dy(x))/(dx) = (x (2+3 x))/(2 y). The Attempt at a Solution I'm not sure how to deal with the derivative being undefined at 0,0 when there are...

Actually, just take an orthogonal coordinate system u,v,w with u perpendicular to x-y. Then the circle defined by (x+y)/2+R(cos(t)*v+sin(t)*w) works, where R=(r^2-(d/2)^2) by the Pythagorean theorem.

I see why it would get me the result, but I have no clue how to prove it. How do we calculate the distance from the origin to that plane?

I am interested in the case 2r>d. The other two I have figured out. It's so obvious but I can't do it rigorously.

I've already solved the case 2r=d. For 2r>d, how do we show |u| cos(theta) = d/2 has infinitely many solutions? Is it just because we can take a ball with radius r around x, and that surface has infinitely many points? That seems wrong... EDIT: Yeah I don't think |u|=d/(wcos(theta)) is a...

That looks really slick. However, I lose you at the key point "Then note that |w| = |x - y| = d and check that there are infinitely many u satisfying this equation if the condition on d is satisfied." Could you explain further? How do we know there are infinitely many solutions? It looks like...