Recent content by Mr.Miyagi

What to read after "Understanding Analysis" I have worked through Abbot's "Understanding Analysis" thoroughly and would like to learn more about the subject. My goal is to gain a good understanding of real and complex analysis and I also want to work my way up to differential geometry. I think...

There may be a relevant identity, but in any case you can write out the first double summation and rearrange the terms in order to get the second equality. (2/2 + 2/3 + ... 2/n) + (2/2 + 2/3 + ... 2/(n-1)) + ... + (2/2) = (n-1)*2/2 + (n-2)*2/3 + ... + 1*2/n All that remains is to...

You haven't mentioned the domain of the functions in (a_n). It should probably not contain 0, as I think the quantity in question doesn't converge for x=0. I haven't checked it, but I assume that y_{n+1}=y_n^2 is correct. This implies that y_{n}=y_1^{2^n}. Moreover, we can calculate that |y_1|<...

When you suspect that a limit does not exist, you can make use of the sequential criterion for limits. The criterion loosely states a limit \lim_{x→c}f(x)=L exists if and only if for every sequence (x_n) that converges to c it follows that \lim_{n→∞}f(x_n)=L. So if you want to prove that the...

There is an identity for sin^nx which transforms it into a sum of regular sines. Perhaps that is a place to start.

\sin x=\frac{e^{ix}-e^{-ix}}{2i} Does that help? edit: Ah, sorry. Didn't see the mangled tex. just a minute.

Alright, that is a nice strategy. But I still have to prove the special case, which I still can't seem to do. I have made the following attempt at proving the general case in one go. Choose c such that a<c<b and A/(c-a)>1. Let us assume for contradiction that there exists an x_0\in [a,c]...

A is indeed a positive constant. Sorry for the ambiguity. Also, I have solved the second problem, so I don't need help with that anymore.

I just wanted to say that I have solved the second problem. By finding the maximum of x^n(1-x) by differentiating it and setting it to zero, we can get the inequality x^n(1-x) < 1/n. I feel a little silly for not considering that before. I am still breaking my brain over the first problem...

The problem statement Let f:[a,b]→\mathbb{R} be differentiable and assume that f(a)=0 and \left|f'(x)\right|\leq A\left|f(x)\right|, x\in [a,b]. Show that f(x)=0,x\in [a,b]. The attempt at a solution It was hinted at that the solution was partly as follows. Let a \leq x_0 \leq b. For all x\in...

Suppose \lim a_n=L. Let a_{n_i} be an arbitrary subsequence. We wish to prove that \lim a_{n_i}=L. We know that \forall \epsilon>0 \exists N such that n\geq N implies |a_n-L|<\epsilon. We need to show that \forall \epsilon>0 \exists I such that i\geq I implies |a_{n_i}-L|<\epsilon. But if...

That is correct: The average speed does go down, but the number of molecules will go up. Perhaps the following will be more illuminating: As it turns out, the pressure on the walls is equal to one third of the kinetic energy density...

I'll try to pitch in, since I vividly remember being confused when I first saw this. When you see ABf, with A and B operators and f some function, remember you first operate with B on f and then you operate with A on the result (which is a product in your case!). It's not Af \cdot Bf edit...

The total pressure will be higher than the initial pressure, but it won't be five times as high. If you don't quite see how this works, think of the following case: suppose you dropped in some molecules with temperature 0K. What would this do? Well, it wouldn't do anything, because a molecule...

Ugh, sorry about that... Is it too late to claim temporary insanity? :rolleyes: Thanks for correcting it so quickly.