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 Mar6 comment Linear transformations such that $A^3=I$ but $A^2\neq I$ Consider rotating a plane by $120^\circ$. Mar5 answered Proof with derivatives (most likely induction) Mar4 awarded Popular Question Mar3 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 $\lim_\limits{h\to 0}(F(x+h)-F(x))/h=f(x)$? Mar3 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. Mar3 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 Mar3 answered Differentiable $N$ times, $f'(x_{0})=\cdots=f^{(N-1)}(x_0)=0$, $f^{(N)}(x_0)>0\implies f$ increasing on $x_0$? Feb18 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? Feb8 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. Feb8 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. Jan21 answered If $3^x$ and $5^x$ are both integers, is $x$ an integer? Jan21 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). Jan21 comment If $3^x$ and $5^x$ are both integers, is $x$ an integer? Jan19 awarded Necromancer Jan9 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. Dec28 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"? Dec28 comment Two questions about divisible A hint on 1) is $\varphi(10)=4$. Dec22 awarded Constituent Dec14 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 $\lim\limits_{k\to\infty} (-1)^k\neq\infty$. Dec12 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 $\Re (p^s+i)^r$.