Recent content by WaterPoloGoat

Ah. I see. Thank you. I'll work on that.

I've started doing a comparison test, but I got a little stuck. I need to play around with it more. I noted that 1/sqrt(x^3+x) is < 1/sqrt(x^3) = x^(-3/2). If the integral x^(-3/2) converges, then so does the original problem. However, I ran into a little snag: The integral of...

Does anyone know of helpful tests that can help me determine whether an improper integral converges or diverges? Specifically ones where you don't have to solve the integral? For example, the problem I'm solving has a very complicated solution to the integral: the problem: integrate...

Oh, my, goodness. I've been trying to ride a bike with no steering. Thank you so much for your help. What a dunce I've been!

Dang it. I keep working and re-working this problem, and x keeps getting eliminated. I've also tried substituting f(y)=f(x+y)/f(x), but this doesn't help one bit. I'll have to think about this. Thank you for your help! I will keep working on it.

f(x)+f(y)=f(x)f(y) f(x)=f(x)f(y)-f(y) f'(x)=limh->0 f(x+h)f(y)-f(y)-[f(x)f(y)-f(y)]/h f'(x)=limh->0 f(h)f(y)/h. Is that correct?

I don't know how I got f'(0)=0. I just jotted it down, but now I realize that's not a valid conclusion.

Would I apply f(x+h) to both sides? if so, I got: f(x+h+y)-f(x+y)=f(x+h)f(y)-f(x)f(y) which simplifies to: f(h)=f(h)f(y) Am I on the right track? What do I do with this?

Given: f(x+y)=f(x)f(y). f'(0) exists. Show that f is differentiable on R. At first, I tried to somehow apply the Mean Value Theorem where f(b)-f(a)=f'(c)(b-a). I ended up lost... Then I tried showing f(0)=1, because f(x-0)=f(x)f(0) and f(x) isn't equal to 0. However, with that...

Homework Statement We know that f is uniformly continuous. For each n in N, we define fn(x)=f(x+1/n) (for all x in R). Show that fn converges uniformly to f. Homework Equations http://en.wikipedia.org/wiki/Uniform_convergenceThe Attempt at a Solution I know that as n approaches infinity...

Homework Statement Define f(x)=sin(x)sin(1/x) if x does not =0, and 0 when x=0. Have to prove that f(x) is continuous at 0. Homework Equations We can use the definition of continuity to prove this, I believe. The Attempt at a Solution I know from previous homework...

snipez, thank you for correcting my obvious error when it comes to problem (b). And I know that the sequence from (c) is bounded, so the Bolzano-Weierstrass theorem will help me there. Do you have any suggestions on how to prove that these other subsequences converge (or don't)?

The Bolzano-Weierstrass Theorem simply states that "every bounded sequence has a convergent subsequence." It doesn't really seem to help here because the sequences themselves don't seem to be bounded.

Homework Statement Which of the following sequences have a convergent subsequence? Why? (a) (-2)n (b) (5+(-1)^n)/(2+3n) (c) 2(-1)n Homework Equations Cauchy Sequence Bolzano-Weirstrass Theorem, etc. The Attempt at a Solution (a) The sequence I get is...