Recent content by Whistlekins

Can you expand on what you mean by maximized? Would that happen when n = 0 and p -> ∞?

Homework Statement Prove that the series \sum_{n=0}^\infty(-1)^n\frac{x^{2n+1}}{2n+1} is well-defined and differentiable on (-1,1). Homework Equations The Attempt at a Solution I know that the function is the series expansion of arctan(x), but that it not we are showing here...

Understood. Now the second part of the question, can I just use the fact that limits of the product of two functions is equal to the product of the limits of the functions, and the fact that g_n(x) = nx^n is a subsequence of h? Is this all that is required for a proof?

I don't understand why I need to take the log. Aren't I changing the function then?

Homework Statement Let (fn) be the sequence of functions defined on [0,1] by fn(x) = nxnlog(x) if x>0 and fn(0)=0. Each fn is continuous on [0,1]. Let a be in (0,1) and ha(y) = yay := yey log(a). Using l'Hospitals rule or otherwise, prove that limy->+∞ ha(y) = 0. Then considering...

If I make A the set of all maxima of all C_n = \{ 2-\frac{1}{2n}\} for all natural numbers n, Then the maximum of this set A does not exist since it is infinite. However the least upper bound would be the limit = 2. Am I correct in thinking this? Then sup(C) = sup(A) = 2. Is this...

Well, now I have no idea. How else could I show that 2 is not in C? Unless 2 IS in C...

I don't really understand what you mean. 2 cannot be in C because 2-1/2n -> 2 as n -> ∞, so C_n = (1,2) as n -> ∞, wouldn't it?

Homework Statement We have C_n = [1-\frac{1}{n},2-\frac{1}{2n}] and C = C_1 \cup C_2 \cup C_3 \cup ... and are asked to describe the interval C and then prove that it is actually what we say it is. Homework Equations The Attempt at a Solution I am guessing that C = [0,2) and...

Ahh I overlooked that, thanks! I think I've got it now.

I have, that's how I arrived at the reduced expression. The questions is where to go from there. I could always use the property that if f and g are continuous at a point x_0\in \mathbb{A} then f+g is continuous at x_0. But I don't know how to prove that cos(x) is continuous on the domain.

Homework Statement We have f(x) = \frac{x^{2}+x-2}{x-1}+cos(x) , x\in\mathbb{R}\setminus \{1\} and wish to prove that it is continuous on its domain. Homework Equations The delta-epsilon definition of the continuity of a function. The Attempt at a Solution I've managed to reduce |f(x) -...

Cool, thanks for helping me clear my confusion. I guess that never really got explained to me by anyone and I never picked up on it.

So if I previously define the domain, I can't change that domain unless I write an entirely new function? Would it be true that if h(x) = x+2 , for all x in R\{1}, then f = h?

I understand that. But why can't I write g(x) = (x^2+x-2)/(x-1) = x+2 ?