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comment Class of matrices for wich $A^T=J-A.$
This condition just fixes the symmetric part of a matrix to $A_S:=\frac{1}{2}(A+A^T)=\frac{1}{2}J$. It implies weaker condition like $\mathrm{tr}(A)=\frac{n}{2}$, and so on.
2d
comment How to calculate the $n$-th member of sequence $a_{n+1}=\sqrt{y+a_{n}}$
What are your apparently missing continuity requirements?
Jun
30
comment Integral of cos(1/x) dx
it returns to the integrand*
Jun
29
comment Proving that $\ln ^3|x|=x$ has exactly 3 real solutions
@Nescio: I'm not sure. You left an hour to answer before it was done - maybe that's the best way if you're sure you're done with the problem. But I think otherwise you may get a more detailed response. I leave question open for one day on principle, just so that the standard userbase gets to see it. If the question pops up with a the green being solved mark, I assume people will tend to not put more though into it. I don't know how it affects the stream of people visiting would criticise the answer or point out errors.
Jun
29
comment Proving that $\ln ^3|x|=x$ has exactly 3 real solutions
Interesting question, the behavior of $\ln(x)^r-x$ for $r\in(2,4)$ is actually somewhat complicated. I'd like to know at which $r$ the number of $x$-intersections in which interval changes. PS: Accepting an answer fast, e.g. within a time where half of the world is sleeping, doesn't help you or people who potentially want to contribute.
Jun
25
comment How do I show :$\sum_{n=0}^{\infty }\frac{(-1)^n}{n+1}=\ln2$?
@RobertIsrael: His answer may lead him straight to his knowledge about a series representation, which he can use.
Jun
25
comment How do I show :$\sum_{n=0}^{\infty }\frac{(-1)^n}{n+1}=\ln2$?
Let me ask you how you'd compute $\ln(37)$.
Jun
25
comment Let $ S=\{(x,y)\in\mathbb{R}^2 \ | \ x^2+y^2=1 \text{ and } y\geq 0\}$. Determine $S+S+…+S $.
Mhm ... that the sum of set without the restriction is the disk with radius $n$ is evident. I'm not how to read your last line as an instruction to prove the restricted case.
Jun
25
revised Let $ S=\{(x,y)\in\mathbb{R}^2 \ | \ x^2+y^2=1 \text{ and } y\geq 0\}$. Determine $S+S+…+S $.
added 139 characters in body
Jun
25
answered Let $ S=\{(x,y)\in\mathbb{R}^2 \ | \ x^2+y^2=1 \text{ and } y\geq 0\}$. Determine $S+S+…+S $.
Jun
25
comment What is the smallest unknown natural number?
@JosephO'Rourke: If it depends on it, say if it's true only if choice holds, I suppose that then means the problem is kicked into the realm of non-constructive statements? That would be a bummer.
Jun
23
comment Proving consistency by constructing models? How and why?
Good answer! The first bullet point raises a question for me, because it doesn't seem to say anything: If someone did come up with an inconsistent set theory, you would just not count it to the "kind of set theory" that "we" came up with. So is there a historical case where some serious post-Russell mathematicians came up with a set theory the we found to be inconsistent, but that was published and survived for a non-trivial time?
Jun
22
comment Compute a complicated integral
You can factor out $\alpha$ via a re-parametrization of $u$. After this, it's a matter of matching the integral to the definition of the Beta function.
Jun
22
comment Prove that I(n) is a strictly decreasing series: I(n)= $\int_0^1 \frac{(x^n)}{(x+1)} dx$
No it doesn't. If $f(x)<g(x)y$ for all $x\in [a,b]$ except for a finite number of points where $f(x)$ is $77$ and $g(x)$ is $0$, then the integral over $g$ is still larger than that over $f$. Isolated points are ignored by the integral.
Jun
22
comment Prove that I(n) is a strictly decreasing series: I(n)= $\int_0^1 \frac{(x^n)}{(x+1)} dx$
The value of the integrand at discrete points like the one at 0 or 1 doesn't matter for the integral. Can you elaborate more what you mean by the comment?
Jun
21
comment Convergence of $\prod\limits_{k=1}^n \left(1+\frac{1}{k^2}\right) $
Accepting an answer after 10 minutes takes the fun out of it.
Jun
21
comment How was the equation re-written?
@Pradeep: I see your question has been answered. A general tip: Write spaces after points. Not "Thank You.But i", but rather "Thank you. But I".
Jun
20
revised How was the equation re-written?
added 21 characters in body
Jun
20
revised How was the equation re-written?
added 264 characters in body
Jun
20
answered How was the equation re-written?