Reputation
Next tag badge:
397/400 score
130/80 answers
Badges
10 125 315
Newest
 Nice Answer
Impact
~3.0m people reached

Apr
28
comment Continuity, algebraic and rational numbers
I think you misunderstand what I had in mind, but it'll be better to edit the answer rather than trying to clarify with further comments. I don't have time right now, but maybe later tonight...
Apr
27
comment Orthonormal basis for Hilbert space
Hint: $$\left\|\sum_{j=n+1}^m b_j/j\right\|^2 = \sum_{j=n+1}^m 1/j^2 < 1/n$$
Apr
27
comment How do I know the probability for me to be ranked in the 2nd place or last 2nd place?
You don't know them, and can't know them unless you assume a particular probability model for the scores.
Apr
27
answered Orthonormal basis for Hilbert space
Apr
27
comment Continuity, algebraic and rational numbers
If $f(w) = z \in \mathbb Q[i]$ with multiplicity $m$, take $r > 0$ such that $w$ is the only zero of $f - z$ within distance $r$ of $w$. By Rouché's theorem there exists $N$ such that for $n > N$, $F_n - z$ has exactly $m$ zeros (counted by multiplicity) within distance $r$ of $w$. But by construction, such zeros (if $n$ is large enough) are in $\mathbb A$ and will also be zeros of $F_k - z$ for all $k > n$, and therefore of $f - z$: thus they must be $w$, and $w \in \mathbb A$.
Apr
27
answered Example of a bounded space which is not totally bounded
Apr
27
answered Minimum and maximum with lagrange multiplier
Apr
27
comment Singular points of $\displaystyle \sin \left( \frac{1}{\cos\frac{1}{z}}\right)$
If you write $1/z_n = \pi/2 + k \pi + s$, $\cos(1/z_n) = (-1)^{k+1} \sin(s)$, and this will be $1/t$ if $s = (-1)^{k+1} \arcsin(1/t)$. As $n \to \infty$ you have $t = (n+1/2) \pi \to \infty$ so $1/t \to 0$ and $s \to 0$, i.e. $1/z_n \to \pi/2 + k \pi$ and $z_n \to 1/(\pi/2 + k\pi)$.
Apr
27
comment Continuity, algebraic and rational numbers
I think we can still have $f^{-1}(\mathbb Q[i]) = \mathbb A$ by modifying the construction, ensuring at stage $j$ that any $w$ with $|w| \le j$ for which $F_j(w)$ is of the first $j$ members of $\mathbb Q[i]$ (in a fixed enumeration) is in $\mathbb A$, and making $f_k(w) = 0$ for $k > j$.
Apr
27
revised Continuity, algebraic and rational numbers
added 1780 characters in body
Apr
27
comment Continuity, algebraic and rational numbers
Bijective on $\mathbb R$, I hope you mean. There aren't too many entire functions bijective on $\mathbb C$.
Apr
27
answered Continuity, algebraic and rational numbers
Apr
27
comment Singular points of $\displaystyle \sin \left( \frac{1}{\cos\frac{1}{z}}\right)$
$\sin(t) = \pm 1$ if $t$ is an odd multiple of $\pi/2$, say $(n+1/2) \pi$. Then you want $t = 1/\cos(1/z_n)$, with $1/z_n$ near $\pi/2 + k \pi$...
Apr
27
answered Singular points of $\displaystyle \sin \left( \frac{1}{\cos\frac{1}{z}}\right)$
Apr
27
comment Does this power series $\sum_{n=0}^{\infty} \frac{n^n}{(n!)^2}x^n$ converge for all $x$?
The ratio test works well here.
Apr
27
comment Multiplying binomials to come up with $ y^8 - 256 $
Right to left may be easiest.
Apr
27
answered Numerical solution of an ODE system of equations using RK4
Apr
27
answered Markov Chain that isn't Irreducible
Apr
27
answered How to find an unknown matrix which when multiplied with a vector gives the cross product of 2 vectors
Apr
27
awarded  functional-equations