User HeroZhang001

8 votes

Use of $\sum_{r=1}^{\infty} \frac{1}{r^2} = \frac{\pi^2}{6} $ in evaluating $I$

answered May 14 at 10:39

8 votes

Accepted

Why $1− \frac12 + \frac13 − \frac14 + \frac15 - \cdots$ is not zero but $\ln2$

answered Jun 1 at 9:40

6 votes

Accepted

Product vs. join of subgroups of a finite group

answered Apr 27 at 15:56

4 votes

Is there a sequence $a_n$ decreasing to $1$ such that $\sum n^{-a_n}$ converges?

answered Apr 28 at 14:03

4 votes

Accepted

Please tell me why my method of solving $\sin\left(\frac{\pi}{3}-2\,x\right)-\sin\left(6\,x\right)=2$ is not correct.

answered Jan 29 at 14:15

3 votes

Let $G$ be a finite group and $\varphi: G \to \varphi(G)$ be a group homomorphism. Prove that $\varphi$ is injective.

answered Feb 5 at 2:43

3 votes

Find the solution of the differential equation $xdx+ydy+\frac{xdy-ydx}{x^2+y^2}=0.$

answered May 1 at 7:08

3 votes

Accepted

Vector fields that make up a local frame are continuous

answered Mar 5 at 8:46

2 votes

Accepted

No Identity Element

answered Feb 20 at 0:36

2 votes

Accepted

Proving that $ 13 | 145^6 + 1$

answered Feb 4 at 3:07

2 votes

Accepted

Why is it wrong to understand derivable and differentiable in binary functions in this way？

answered Dec 5, 2022 at 15:18

2 votes

Prove that $\lim _{h \rightarrow 0} \frac{f(2 h)-f(-5 h)-21 h}{h^2} < \infty$ for a $C^2(\Bbb R)$ function $f$ with $f(0)=1,f'(0)=3$, and $f''(0)=5$

answered Jan 28 at 13:46

1 vote

Accepted

To prove the group of homeomorphisms is a metric space

answered Jan 29 at 7:06

1 vote

Accepted

Semifinite Measure

answered Jan 29 at 7:47

1 vote

Simplify $(p \vee q) \wedge(p \vee \neg q) $ to obtain $ (p \vee(q \wedge \neg q))$

answered Jan 29 at 13:38

1 vote

Accepted

Calculate the sum: $\sum_{n=1}^\infty \frac{1}{n^3+3n^2+2n}$

answered Dec 14, 2022 at 17:09

1 vote

Uniform continuity of the function $f(x)=x^a$

answered Dec 18, 2022 at 16:24

1 vote

Proof of $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}$

answered Apr 19 at 13:56

1 vote

Accepted

How is $\frac{1}{2\pi} \int_{-\pi}^{\pi} e^{in\theta}e^{-ik\theta} d\theta = \delta_{n, k}$?

answered Feb 7 at 0:49

1 vote

Why does $\phi_[X,Y]=[\phi_X,\phi_*Y]$?

answered Feb 20 at 7:23

1 vote

Accepted

Find the volume of $\mathbb S^n$ (the surface area of $(n+1)$-dimensional unit ball) by using differential geometry

answered Mar 14 at 6:33

1 vote

Accepted

Improper integral of Euler's constant and trigonometric

answered Mar 16 at 1:30

1 vote

Mathematical Analysis I by Zorich: show $f^{-1}(f(A)) \supset A$, $f(f^{-1}(B^{'})) \subset B'$

answered Jun 18 at 13:48

0 votes

Accepted

Is $\{(\mathbb R,\phi)\}$ a smooth atlas for $\mathbb R$, where $\phi(x)=\left\{ \begin{array}{ll} x & x<0, \\ 2x& x\geq0~? \end{array} \right.$

answered Mar 3 at 2:15

0 votes

Accepted

How do we show a matrix is similar?

answered Feb 14 at 7:55

0 votes

Show the measure $\overline{\mu}$ is complete

answered Feb 6 at 7:47

0 votes

Equality condition in Minkowski's inequality for $L^{\infty}$

answered Feb 23 at 10:46

0 votes

Accepted

$\sum_{k=1}^{n} \sin (kx)$

answered Dec 19, 2022 at 14:16

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28 Answers

Use of $\sum_{r=1}^{\infty} \frac{1}{r^2} = \frac{\pi^2}{6} $ in evaluating $I$

Why $1− \frac12 + \frac13 − \frac14 + \frac15 - \cdots$ is not zero but $\ln2$

Product vs. join of subgroups of a finite group

Is there a sequence $a_n$ decreasing to $1$ such that $\sum n^{-a_n}$ converges?

Please tell me why my method of solving $\sin\left(\frac{\pi}{3}-2\,x\right)-\sin\left(6\,x\right)=2$ is not correct.

Let $G$ be a finite group and $\varphi: G \to \varphi(G)$ be a group homomorphism. Prove that $\varphi$ is injective.

Find the solution of the differential equation $xdx+ydy+\frac{xdy-ydx}{x^2+y^2}=0.$

Vector fields that make up a local frame are continuous

No Identity Element

Proving that $ 13 | 145^6 + 1$

Why is it wrong to understand derivable and differentiable in binary functions in this way？

Prove that $\lim _{h \rightarrow 0} \frac{f(2 h)-f(-5 h)-21 h}{h^2} < \infty$ for a $C^2(\Bbb R)$ function $f$ with $f(0)=1,f'(0)=3$, and $f''(0)=5$

To prove the group of homeomorphisms is a metric space

Semifinite Measure

Simplify $(p \vee q) \wedge(p \vee \neg q) $ to obtain $ (p \vee(q \wedge \neg q))$

Calculate the sum: $\sum_{n=1}^\infty \frac{1}{n^3+3n^2+2n}$

Uniform continuity of the function $f(x)=x^a$

Proof of $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}$

How is $\frac{1}{2\pi} \int_{-\pi}^{\pi} e^{in\theta}e^{-ik\theta} d\theta = \delta_{n, k}$?

Why does $\phi_[X,Y]=[\phi_X,\phi_*Y]$?

Find the volume of $\mathbb S^n$ (the surface area of $(n+1)$-dimensional unit ball) by using differential geometry

Improper integral of Euler's constant and trigonometric

Mathematical Analysis I by Zorich: show $f^{-1}(f(A)) \supset A$, $f(f^{-1}(B^{'})) \subset B'$

Is $\{(\mathbb R,\phi)\}$ a smooth atlas for $\mathbb R$, where $\phi(x)=\left\{ \begin{array}{ll} x & x<0, \\ 2x& x\geq0~? \end{array} \right.$

How do we show a matrix is similar?

Show the measure $\overline{\mu}$ is complete

Equality condition in Minkowski's inequality for $L^{\infty}$

$\sum_{k=1}^{n} \sin (kx)$