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Nov
16
comment For which positive integers k is this series convergent: $\sum_{n=1}^{\infty}\frac{(n!)^2}{(kn)!}$
The case $k=2$ is not settled using the ratio test, so it needs a different approach. Have you at least tried running numerical tests with specific values of $k$ to get a feel for how the terms in the series decay?
Nov
15
revised Clarification of proof on the completion of a metric space using Cauchy sequences
added 1 character in body; edited title
Nov
15
revised The proof of Cantor's Intersection Theorem on nested compact sets
edited title
Nov
15
revised Question about the Ascoli-Arzelá Theorem proof
edited title
Nov
14
asked The convention for speakers to refer to themselves at the board with a single initial
Nov
13
comment Is $ G \cong G/N \times N$?
If $N$ is a normal subgroup of a finite group $G$ and the order $|N|$ and index $[G:N] = |G|/|N|$ are relatively prime then $G$ is isomorphic not necessarily to the direct product of $N$ and $G/N$, but to some semidirect product of $N$ and $G/N$. This is called the Schur--Zassenhaus theorem.
Nov
12
comment Who first proved Fermat's Last Theorem for polynomials and when?
It goes back to the 19th century. See references on the page math.niu.edu/~rusin/known-math/98/flt_poly, especially the last chapter of Ribenboim's 13 Lectures on Fermat's Last Theorem.
Nov
12
comment Degree 4 extension of $\mathbb {Q}$ with no intermediate field
See Example 4.16 and Remark 4.18 in math.uconn.edu/~kconrad/blurbs/galoistheory/galoisaspermgp.pdf
Nov
10
comment $u_{n+2}-u_{n+1}+2u_n=0$ implies that $\vert u_n \vert\rightarrow +\infty$.
The general formula for a solution of that linear recursion is $u_n = a((1+\sqrt{-7})/2)^n + b((1 - \sqrt{-7})/2)^n$ for some $a$ and $b$. By using Skolem's $p$-adic method, which amounts to viewing $u_n$ as a $p$-adic locally analytic function of $n$ for a suitable prime $p$ (any $p \not= 2$ will work), this sequence provably can't take any value infinitely often unless it is a constant sequence (like all terms equal to 0, as David H points out). Therefore if the $u_n$'s are all integers (equivalently, if $u_0$ and $u_1$ are integers), the magnitude $|u_n|$ must tend to $\infty$ with $n$.
Nov
10
comment Inner product space as dot product
For example, the dot product on $F^n$ is not "positive-definite" when $F$ is a finite field: there is no notion of positivity!
Nov
10
comment Inner product space as dot product
Inner products are defined on vector spaces over the real or complex numbers (or fields sufficiently close to them in structure that the definition carries over more or less directly), but not all vector spaces in math are defined over such fields, e.g., think about vector spaces over finite fields.
Nov
10
comment Inner product space as dot product
The answer is "yes" when the vector space has scalars from the real numbers, but of course not over a general field.
Nov
10
awarded  Good Question
Nov
9
comment Showing that the integral remainder of the Taylor expansion of $f(x)=-\log(1-x)$ goes to $0$
Both of us are using an integral formula for the remainder for essentially the same function at the same point ($-\log(1-x)$ at $x = 0$ for you, $\log(1+x)$ at $x=0$ for me). In your integral formula for $R_n(x)$, make the change of variables $u = (x-t)/(1-t)$. Check that $dt/(1-t) = du/(u-1)$, so after changing variables from $t$ to $u$ your formula becomes $R_n(x) = \int_0^x (u^{n-1}/(1-u))\,du$, which is the integral formula for the remainder in (2.4) of the file I linked to (with an index shift by 1). That this integral tends to 0 is shown in section 3 of my file. It is the same goal.
Nov
8
comment Showing that the integral remainder of the Taylor expansion of $f(x)=-\log(1-x)$ goes to $0$
Your series at the end should start with $k = 1$, not $k = 0$. For a proof that the remainder goes to $0$ (using a different integral formula for the remainder than what you write), see math.uconn.edu/~kconrad/blurbs/analysis/logarctan.pdf.
Nov
7
comment About motivation of Lang's Proof $S_n$ is not solvable for $n\geq 5$.
The 3-cycles generate $A_n$, so it's natural to consider them when you're trying to analyze the normal subgroups of $S_n$. As soon as you have all the 3-cycles in a subgroup of $S_n$ then you have at least half the group in your subgroup.
Nov
7
revised Examples of mathematical discoveries which were kept as a secret
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Nov
7
comment Examples of mathematical discoveries which were kept as a secret
@ypercube: There is no written record of Fermat's reasoning except for a proof that 1 is not a congruent number. See Weil's "Number Theory: An Approach Through History from Hammurapi to Legendre." In the chapter on Fermat, Weil writes often about Fermat being cryptic in his letters. As for the impossibility of solving $x^4 + y^4 = z^4$ in positive integers, that is a consequence of the impossibility of solving $x^4 + y^2 = z^4$ in positive integers, and that follows from 1 not being a congruent number, so in a sense Fermat did leave to posterity a proof of his last theorem for exponent 4.
Nov
7
answered Examples of mathematical discoveries which were kept as a secret
Nov
6
comment Importance of Taylor polynomials
What kind of course is this: high school, college, something else?