The Question : 104 people think this question is useful Moderator Notice: I am unilaterally closing this question for three reasons. The discussion here has turned too chatty and not suitable for the MSE framework. Given the recent pre-print of T. Tao (see also the blog-post here), the continued usefulness of this question is diminished.
The Question : 138 people think this question is useful As a former physics major, I did a lot of (seemingly sloppy) calculus using the notion of infinitesimals. Recently I heard that there is a branch of math called non-standard analysis that provides some formalism to this type of calculus. So, do you guys think
The Question : 150 people think this question is useful I am trying to find a closed form for $$\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx = 0.094561677526995723016 \cdots$$ It seems that the answer is $$\frac{\pi^2}{12}\left( 1-\sqrt{3}\right)+\log(2) \log \left(1+\sqrt{3} \right)$$ Mathematica is unable to give a closed form for the indefinite integral. How can we prove this
The Question : 157 people think this question is useful Let’s say I have $\int_{0}^{\infty}\sum_{n = 0}^{\infty} f_{n}(x)\, dx$ with $f_{n}(x)$ being continuous functions. When can we interchange the integral and summation? Is $f_{n}(x) \geq 0$ for all $x$ and for all $n$ sufficient? How about when $\sum f_{n}(x)$ converges absolutely? If so why? The
The Question : 159 people think this question is useful Suppose you’re trying to teach analysis to a stubborn algebraist who refuses to acknowledge the existence of any characteristic $0$ field other than $\mathbb{Q}$. How ugly are things going to get for him? The algebraist argues that the real numbers are a silly construction because
The Question : 178 people think this question is useful I’ve read many times that ‘compactness’ is such an extremely important and useful concept, though it’s still not very apparent why. The only theorems I’ve seen concerning it are the Heine-Borel theorem, and a proof continuous functions on R from closed subintervals of R are
The Question : 194 people think this question is useful In an abstract algebra class, one learns about groups, rings, and fields, and (perhaps naively) conceives of a modern-day algebraist as someone who studies these sorts of structures. One learns about the classification of finite simple groups, and gains some slight sense of what a
The Question : 405 people think this question is useful $\displaystyle\sum_{n=1}^\infty \frac{1}{n^s}$ only converges to $\zeta(s)$ if $\text{Re}(s) > 1$. Why should analytically continuing to $\zeta(-1)$ give the right answer? The Question Comments : It’s not clear (to me) what you mean by “the right answer”. Since this was migrated from physics, I suspect that
The Question : 771 people think this question is useful What is wrong with this proof? Next to 180 it says “deg” So a whole side is of length $2\cdot \left(\frac{1}{2}\tan{\frac{180^\circ}{n}}\right)$ There are $n$ sides, so it has perimeter ${n\tan{\frac{180^\circ}{n}}}$ The folding method (details at the end of the answer) preserves perimeter. So the polygon’s
The Question : 52 people think this question is useful Suppose f(m,n) is a double sequence in $\mathbb R$. Under what condition do we have $\lim\limits_{n\to\infty}\sum\limits_{m=1}^\infty f(m,n)=\sum\limits_{m=1}^\infty \lim\limits_{n\to\infty} f(m,n)$? Thanks! The Question Comments : see jstor.org/stable/1967602?seq=1#page_scan_tab_contents see en.wikipedia.org/wiki/Tannery%27s_theorem The Answer 1 56 people think this answer is useful A fairly general set of conditions, sufficient
The Question : 51 people think this question is useful On the answers of the question Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio? it was told that $\frac{dy}{dx}$ cannot be seen as a quotient, even though it looks like a fraction. My question is: does $dxdy$ in the double integral represent a multiplication of differentials? The problem
The Question : 1074 people think this question is useful In the book Thomas’s Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $dy/dx$ is not a ratio. Couldn’t it be interpreted as a ratio, because according to the formula $dy = f'(x)dx$ we are able to plug in values for
