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$Some high dimensional geometric weirdness that we can explore with integration:\\ \\ a) By analogy with the volumes by slices of MX2, find an expression for the volume of the n-dimensional unit ball. Express your answer in terms of factorials or the Gamma function. By using Stirling's formula...

\int \frac{e^x-e^{-x}}{e^{2x}+1}\, dx = \int \frac{u^2-1}{u^2(u^2+1)}\, du= \int \frac{2}{u^2+1}-\frac{1}{u^2}\, du\\ \\ = 2\arctan(e^x)+e^{-x}+C\\ \\ $(where $u:=e^x$)$

Oh didn't see you edited it, was the divergence thing always there and I just blind before? In any case, we can still represent the quadratic form by a symmetric, hence diagonalisable matrix Q. If the quadratic form does not represent an ellipse, then at least one eigenvalue is non-positive...

$Let $Q$ be the symmetric matrix associated to the quadratic form defining the conic. $Q$ yields an ellipse iff it is a positive-definite matrix. Standard spectral theory shows that $Q$ is diagonalisable by orthogonal matrices. The resulting matrix is $\textrm{diag}(\lambda_j)$, so after our...

I=\int_0^{2\pi} (\frac{e^{it}+e^{-it}}{2})^{2n}\, dt \\ \\ = 2^{-2n}\binom{2n}{n}\int_0^{2\pi}\, dt\\ \\=\frac{\pi}{2^{2n-1}}\binom{2n}{n}.\\ \\ $(all nonconstant trig monomials integrate to zero.)$

Fair enough, seems a rather subjective matter...to me polylog and polygamma manipulations are okay, but I also have no qualms about using contour integration etc.

Eh I wouldn't say that necessarily, I did the integral in a not particularly imaginative way...there could be a clever shortcut. To me it does seem like computing this integral is roughly the same mathematical "depth" as computing some particular values of the di/trilogarithm though. Anyway...

^ That was good practice! I need to go out for dinner v soon, so will supply gory details later. In short, the answer can be expressed solely in terms of Apery's constant zeta(3). 1. IBP to essentially reduce the integral to log^2(1+x)/x. (The bulk of the difficulty of the originally posted...

Gg man, pretty well played. Shame about that pawn blunder.

Someone has joined, but just isn't moving lol.

Coming? Otherwise someone else join pls :).

https://lichess.org/dhv78LH8

Sweet, will host up in a sec. 15+10 or thereabouts seem decent? or would you prefer something else?

Cool, should be a decent game then. Free now?

Sure I'll be free later this arvo most likely for a few on lichess. How strong are you btw? Also theres some overlap between the part of the brain used in maths and chess respectively imo, but its hard to draw conclusions. The average mathematician who tries his hand at chess tends to do better...

Good bump. People hit me up for games, am really enjoying chess atm :).

I think the Q16h solution is still not quite correct. Whilst it is true that x-k\alpha =< x for each x, we cannot simply bound the product of factors of this form by the product of the x's, as some of these terms will in general be negative. Another way of seeing this problem is that if we...

What is the reasoning behind the second line of the estimate for |Q(k*alpha)| in the solution to Q16h) ? As written, that does not seem complete to me.

Defs try other methods first. It's extremely rare that an inequality will be easier / more natural to prove by contradiction than directly.

What is this claim based on? And ANY heterosexual couples have the legal right to a marriage, and the benefits that come with. There are plenty of heterosexual couples that are in high likelihood going to be awful parents to any children they raise or do not want to have children at all, yet a...