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 14h comment Are there any ways we can determine whether the $\Xi_x$-classes of natural numbers upto $\frac{1}{2}p^2_x -2$ exvert all non-trivial $\Xi_x$-classes? Given your aside about whether the value of $\langle 0, 0, \ldots, 1, \ldots, 0\rangle$ is always $P_x\#/p_y$, I would strongly suggest that you start with the Chinese Remainder Theorem - I haven't dug deeply enough through your notation to wholly understand the questions that you're asking, but I strongly suspect that the CRT will go a long way towards either answering them or leading you to better formulations. 18h comment Prove that the series $\sum_{k=0}^\infty (-1)^k\ \left(\sqrt{k+1}-\sqrt{k}\right)$ converges but not absolutely. Another simple way of showing divergence of the absolute-value series without telescoping: $(\sqrt{k+1}-\sqrt{k})(\sqrt{k+1}+\sqrt{k}) = (k+1)-k = 1$, so $\sqrt{k+1}-\sqrt{k} = \frac{1}{\sqrt{k+1}+\sqrt{k}}$ $\geq\frac{1}{2\sqrt{k+1}}$. But now the sum of the latter diverges by comparison with e.g. the harmonic series. Feb 9 comment Length of a Chord of a circle You actually need radius $x$ - the two intersection points won't form a diameter of the (second) circle, so they won't be distance $(x/2)+(x/2)=x$ apart. Feb 9 comment Extracting the Axis a Quaternion is rotating around from the Quaternion itself Directly Simply normalize the 'purely imaginary' (XYZ) part of the quaternion and that will return the axis. Feb 8 comment If $ax^2+2hxy+by^2+2gx+2fy+c=0$, prove that While this is likely to get a little bogged down in calculation (much like the other approaches), I would consider first rotating to a coordinate system that's missing the cross term (since this won't change the distance but may make some of the subsequent calculations easier). Feb 6 comment How to solve a quadratic inequality that acts like a quadratic equality? Your solutions of your quadratic are wrong, too, on multiple counts. First of all, quadratic roots come in pairs; you can't have a (rational) quadratic with one root rational and the other irrational. Secondly, all the coefficients of your quadratic are positive, so any real roots have to be negative - if $t\geq 0$ then $t^2\geq 0$ and so $3t^2+4t+1\geq 1$. Feb 5 comment A formula for length of representation of a number in a “base” without zeros A broad hint: the transitions come at $3, 3+3^2, 3+3^2+3^3, etc.$ You should be able to find a closed form for these sums (the geometric series will be a help); then you can use algebra to find a proper inversion of it. Feb 5 comment A formula for length of representation of a number in a “base” without zeros What have you tried? Have you noticed where the 'switch-over' points are? That may help you craft a formula similar to the one you know for two items... Feb 4 comment Why doesn't the limit $\lim_{(x,y) \rightarrow (0,0)} \frac{ e^{x+y} - x - y}{\sqrt{x^2 + y^2}}$ exist? This should probably be a comment on the original question. Feb 4 comment Why doesn't the limit $\lim_{(x,y) \rightarrow (0,0)} \frac{ e^{x+y} - x - y}{\sqrt{x^2 + y^2}}$ exist? How do you get your second equality? It looks like you're assigning a value to $\lim_{r\to0}e^{kr}/r$... Feb 4 comment Prove $\int x^n\,dx=\frac{x^{n+1}}{n+1}.$ What's the context of this question? What have you learned to this point? It's surely more than just the 'area under the curve' definition - do you know anything about Riemann Sums, for instance? It's not too hard to use some basic combinatorics to show that $\sum_{i\leq x} i^n = \frac{x^{n+1}}{n+1}+O(x^n)$ and that's actually enough to prove the result by working with a Riemann sum definition, but without knowing more about your knowledge it's all but impossible to unpack that into a specifically meaningful proof. Feb 3 comment Does $\sum_{i=1}^n \frac{1}{i^2}=O(\ln(n))$? Minor nitpickery (that should also go on the other post, but): it's IMHO much more pedagogically useful to think of $O(f(n))$ as a set of functions and then to write, for instance, $\sum_{i=1}^n \frac1{i^2}\in O(\ln n)$ - that is, that it's a member of the set - rather than saying that it equals $O(\ln n)$. For instance, this lets you straightforwardly express the stronger result that people are talking about: $\sum_{i=1}^n\frac1{i^2}\in O(1)\subseteq O(\ln n)$, so the sum must also be $\in O(\ln n)$. Feb 1 comment Confusion regarding $\log(x)$ and $\ln(x)$ @EricLippert I disagree - when the base of a logarithm is relevant (e.g., when constant factors matter and one isn't just saying something like $\Theta(n\log n)$ ) base-two logs are generally written as $\lg$ rather than $\log$. More often than not, though, results are talked about 'up to a constant factor' asymptotically and so the base of a logarithm is moot. Feb 1 comment Proof that $\sum_{i=0}^n {n\choose i}2^{i}=3^{n}$ (Note too that this instantly generalizes to the full binomial theorem, using $m+n$ ordered parts rather than just $1+2$) Jan 31 comment Example of set with irrational upper arithmetic density? Generically, the set of numbers of the form $\lfloor n\alpha\rfloor$ for $\alpha\gt 1$ will have density $\frac1\alpha$; see en.wikipedia.org/wiki/Beatty_sequence for more details on this. Jan 31 comment Example of set with irrational upper arithmetic density? Are you familiar with the (infinite) Fibonacci Word? Jan 31 comment Show that $X(X'X)^{-1}X$ is identical to $XP(P'X'XP)^{-1}P'X'$ @Clarinetist if that isn't the case, neither formula is defined. Jan 31 comment Show that $X(X'X)^{-1}X$ is identical to $XP(P'X'XP)^{-1}P'X'$ (Take $A=P', B=X'X,C=P$) Jan 31 comment Show that $X(X'X)^{-1}X$ is identical to $XP(P'X'XP)^{-1}P'X'$ If $P$ is an invertible mattix then this is trivial using the formula $(ABC )^{-1} =C^{-1}B^{-1}A^{-1}$. Jan 30 comment Is there a better solution than $\int_{1}^{\ln 2} \frac{e^x\,dx}{1 +e^{2x}} = \arctan(2) - \arctan(e)$ How do you define a 'non-trigonometrical answer' when your final form contains an innately trigonometric function?