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Mar
6
comment Linear transformations such that $A^3=I$ but $A^2\neq I$
Consider rotating a plane by $120^\circ$.
Mar
5
answered Proof with derivatives (most likely induction)
Mar
4
awarded  Popular Question
Mar
3
comment An attempt at an epsilon-delta proof of the fundamental theorem of calculus
"...we know that $\exists \delta >0$ such that for any $\varepsilon >0$..." You mean the contrary logical order: for any $\varepsilon$ there is a $\delta$. That said, can you show that $\lim_\limits{h\to 0^+} (F(x+h)-F(x))/h=\lim_\limits{h\to 0^-}(F(x+h)-F(x))/h=f(x)$ implies $\lim_\limits{h\to 0}(F(x+h)-F(x))/h=f(x)$?
Mar
3
comment Differentiable $N$ times, $f'(x_{0})=\cdots=f^{(N-1)}(x_0)=0$, $f^{(N)}(x_0)>0\implies f$ increasing on $x_0$?
The remainder is usually written $r_n(h)$ with $\lim_\limits{h\to 0}r_n(h)/h^n = 0$; here I wrote $\rho(h) = r_n(h)/h^n$ with $\lim_\limits{h\to 0} \rho(h) =0$ and simplified the expansion for our analysis.
Mar
3
revised Differentiable $N$ times, $f'(x_{0})=\cdots=f^{(N-1)}(x_0)=0$, $f^{(N)}(x_0)>0\implies f$ increasing on $x_0$?
added 16 characters in body
Mar
3
answered Differentiable $N$ times, $f'(x_{0})=\cdots=f^{(N-1)}(x_0)=0$, $f^{(N)}(x_0)>0\implies f$ increasing on $x_0$?
Feb
18
comment Prove that $a_1^2+a_2^2+\dots+a_n^2\ge\frac14\left(1+\frac12+\dots+\frac1n\right)$
"(...) which means that $a_k\geq \sqrt{k}-\sqrt{k-1}.$" I don't follow, is the sum bounded above by something?
Feb
8
comment supremum and infimum: $\frac{n^n}{n!^2}$
This is off by a factor of $n+1$ ($a_{n+1}$ is $(n+1)^{n+1}/(n+1)!^2$, not $(n+1)^{n}/(n+1)!^2$). Of course, the answer is correct despite this.
Feb
8
comment supremum and infimum: $\frac{n^n}{n!^2}$
You will find the ratio $a_{n+1}/a_n$ very interesting. Use it to show that $a_{n+1}/a_n <1/2$ from some point on.
Jan
21
answered If $3^x$ and $5^x$ are both integers, is $x$ an integer?
Jan
21
comment The convergence of $\sum_{n=2}^{\infty} {1\over \ln(n!)}$
The divergence of the $1/n\log n$ series can be seen with Cauchy's condensation test (which can be proven with elementary tools).
Jan
21
comment If $3^x$ and $5^x$ are both integers, is $x$ an integer?
Related: mathoverflow.net/questions/17560/…
Jan
19
awarded  Necromancer
Jan
9
comment Mistake in Spivak's *Calculus on Manifolds*?
There is indeed a mistake in this exercise. If I recall correctly, he originally meant $\lambda_i$ instead of $|\lambda_i|$. This is a related question.
Dec
28
comment Relation between norm and determinant of a linear operator
Saying the obvious, the matrix $\begin{pmatrix} 1 & -1 \\ 1 & 0 \end{pmatrix}$ has determinant $1$ and it does not preserve norms as the identity matrix does. What do you mean by "present the norm as the determinant"?
Dec
28
comment Two questions about divisible
A hint on 1) is $\varphi(10)=4$.
Dec
22
awarded  Constituent
Dec
14
comment Question about proving $\displaystyle\lim_{n\to\infty} n=\infty$ using the limit definition for a converging sequence
This argument only shows the limit is not converging to a real number. The sequence $(1,-1,1,-1,...)$ also does not converge to any real number, and $\lim\limits_{k\to\infty} (-1)^k\neq\infty$.
Dec
12
comment How do I prove that $x^s=(-1)^k \sum_{k=0}^{(r-1)/2}\binom{r}{2k}p^{s(r-2k)}$ has no solutions?
The sum is $\Re (p^s+i)^r$.