Our specification only has these points listed for it i.imgur.com/ZjTnoRR.png and the answers to some of the past questions seem intuitive so I don't think it goes into much depth
@AlexanderGruber You don't believe in infinite things; I don't believe in primes $p$ such that there are infinitely many primes $q$ with $p \equiv 1 \mod q, q \equiv 1 \mod p^n$ for some $n$
@AlexanderGruber that's fine, then. ZF {minus infinity} is a consistent theory.
@AlexanderGruber "does anybody know what the status of the whole π+e thing is? Is there a particular thing that is difficult about it, or do we just not really know what angle to attack the problem from"
it's very hard in general to prove that numbers are irrational or transcendental unless they have a good reason to be. the reason we can do it for $\pi$ and $e$ is that they have some very nice functions that they matter to (trig functions for the first; the latter has a quickly converging power series, showing it's very well-approximable by rationals)
there's nothing like this, as far as we know, for $\pi + e$
so we don't really have any good techniques to attack them with.
I deleted that because he told me recently that he thinks math is a solitary endeavor and does not need to be communicated, so I've decided that I should not encourage people to communicate with him
Hi guys, I have a question from my old test that I need clarification on. Is this something suited for main? I'd post the question and post the solution and then ask for clarification. What would the title be of such a post?
@Mike: That's absurd. There's way more geometry in K-N than in Peter's book. Perhaps a bit less analysis and definitely more algebra. The subject has changed in 50 years.
@Ted He wasn't actually hostile, of course. The joke was that it's about general affine connections, which he has so far, I think, found little use for in his work.
(This was friendly banter... I guess that wasn't clear.)
Hi, I have problems with the following I'd appreciate any hint. Let $0\le f_n \uparow f$, so there is a sequence of simple function for each $n$ s.t., $0\le f_{n,\ell}\uparrow f$. If we define $g n=\text{max}\{f_{1,n}\ldots f_{n,n}\}$, it is clear that $g_n$ is simple, non-negative and nondecrescent, but I have problems to show that $g_n\uparrow f$, I thought it was simple but I find problems in the argument. I'd appreciate any suggestion. Thanks
I don't know why I have problems to show the convergence, intuitively is clear that the $g_n\uparrow f$. But I have troubles to develop the argument formally.
@MikeMiller: working at a software company in the meanwhile, and trying to learn a lot math when I go home.
Many excellent places. To save the (excessively) long list, UCLA, Michigan, UC Berkeley, Brown, Northwestern, Yale, NYU, Columbia are some of the usual suspects.
So, I was given this really nasty problem to solve
Suppose C is parametrized by
$\mathbf{g}(t) = \left[\begin{array}{c}e^{t^{3}\cos\!\left(2\pi t^{25}\right)}\cr
t^{6}+3t^{3}+3\cr
e^{\sin\!\left(3t\right)}\!\left(t^{3}-t\right)\cr\end{array}\right] \ , \quad 0\le t\le 1 \ .$
Let $\ome...
While thinking about this post of Clin, I wanted to consider the polynomial: $P(z) = 1+x_1z+x_2z^2+\cdots+x_nz^n$, satisfying:
(I) $1\geq x_{1}\geq x_2\geq\cdots\geq x_{n}\geq0$ and $\sum\limits_{k=1}^{n}x_{k}=1$.
Then, $P(1) = 2$ and $P$ has no root (real or complex) inside the disc $|z| < 1$...
What tools would you recommend me for computing the limit below?
$$\lim_{n\to\infty} \frac{\sqrt{n}}{4^{n}}\sum_{k=1}^{n}\frac{\displaystyle \binom{2n-1}{n-k}}{(2k-1)^2+\pi^2}$$
As soon as any useful idea comes to mind I'll make the proper update with the new findings.
I'm trying to find a closed form for the following sum
$$\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n},$$
where $H_n=\displaystyle\sum_{k=1}^n\frac{1}{k}$ is a harmonic number.
Could you help me with it?
@DanielFischer Oh I see, it's because it is a fixed $\delta$ value in this case so the sum will converge to $1$ in $\frac{1}{\delta}$ steps of the series.
It's trivial to notice by Cauchy-D'Alembert criterion that
$$\lim_{n\to\infty} \frac{n}{\sqrt[n]{n!}}=e \Rightarrow \frac{1}{\sqrt[n]{n!}}\approx \frac{e}{n}$$
and hence the conclusion.
Q.E.D.
@JasperLoy Yeah, the surprises come from people you never expect ... (it wasn't my case here since in the past when I quarreled with him, he began to downvote me - and he also has a downvoting pattern)
Hello everyone! To show that an ideal I=<f(x)> (where f(x) is irreducible) is a maximal ideal of K[x] do we have to show that K is a field and that I=<f(x)> a prime ideal?
@UserX: At the high school where I'm teaching, in the second last year (where they typically learn things like differentiation and integration), students have to choose from a range of optional courses. There are courses in all kinds of subjects, e.g. literature, economics, security, physics, chemistry etc, and I'm offering a course in mathematics. I was told usually only very few students enrol in maths courses so they won't take place (minimum amount of students).
@UserX: Every teacher is allowed to describe the course in a few lines of text, but nothing more. And usually, students will learn some basic probability in high school.
What tools would you recommend me for computing the limit below?
$$\lim_{n\to\infty} \frac{\sqrt{n}}{4^{n}}\sum_{k=1}^{n}\frac{\displaystyle \binom{2n-1}{n-k}}{(2k-1)^2+\pi^2}$$
As soon as any useful idea comes to mind I'll make the proper update with the new findings.
@TheArtist after a while you reach the second formula here. Well, not the way you expect. The idea is to differentiate once with respect to $z$ and then after you set $z=1$ you get in the right side the integral you lastly obtain when using the way I told you about above.
In the following link http://integralsandseries.prophpbb.com/topic407-50.htmlI found
the integral & the evaluation of
$$\displaystyle \int^1_0 \frac{\text{Li}_2(x)^3}{x}\,dx$$
I'll also include a simpler version together with the question: is it possible to find some easy
ways of computing bo...
@Chris'ssis lul ,.. those double euler sums are piece of cake ! (wonder why did S go through all the trouble of writing them like that!) what kind of alternative approach are you looking for ? :-)
In the following link here I found the integral & the evaluation of
$$\displaystyle \int^1_0 \frac{\text{Li}_2(x)^3}{x}\,dx$$
I'll also include a simpler version together with the question: is it possible to find some easy
ways of computing both integrals without using complicated sums that r...
@Studentmath You have $\lvert f(x) - f(y)\rvert \leqslant \lvert f(x) - f(z)\rvert + \lvert f(z) - f(y)\rvert$ for all functions $f$ (and all $x,y,z$ in its domain). That's just the triangle inequality of $\lvert\,\cdot\,\rvert$.
@DanielFischer So, can I just say that , since $X^2 - 1 = (X-1)(X+1)$ is a multiple of $X-1$, we have that: $\langle X-1, Y + X^2-1\rangle = \langle X-1,Y\rangle$ ? Or do I have to explain it further?
So, I was given this really nasty problem to solve
Suppose C is parametrized by
$\mathbf{g}(t) = \left[\begin{array}{c}e^{t^{3}\cos\!\left(2\pi t^{25}\right)}\cr
t^{6}+3t^{3}+3\cr
e^{\sin\!\left(3t\right)}\!\left(t^{3}-t\right)\cr\end{array}\right] \ , \quad 0\le t\le 1 \ .$
Let $\ome...
While thinking about this post of Clin, I wanted to consider the polynomial: $P(z) = 1+x_1z+x_2z^2+\cdots+x_nz^n$, satisfying:
(I) $1\geq x_{1}\geq x_2\geq\cdots\geq x_{n}\geq0$ and $\sum\limits_{k=1}^{n}x_{k}=1$.
Then, $P(1) = 2$ and $P$ has no root (real or complex) inside the disc $|z| < 1$...
What tools would you recommend me for computing the limit below?
$$\lim_{n\to\infty} \frac{\sqrt{n}}{4^{n}}\sum_{k=1}^{n}\frac{\displaystyle \binom{2n-1}{n-k}}{(2k-1)^2+\pi^2}$$
As soon as any useful idea comes to mind I'll make the proper update with the new findings.
I'm trying to find a closed form for the following sum
$$\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n},$$
where $H_n=\displaystyle\sum_{k=1}^n\frac{1}{k}$ is a harmonic number.
Could you help me with it?
