User HGF

20 votes

5 answers

784 views

Find the limit $\lim\limits_{s\to0^+}\sum_{n=1}^\infty\frac{\sin n}{n^s}$

asked Mar 19, 2022 at 4:10

9 votes

1 answer

204 views

$\sum_{n=1}^\infty\ln(1+u_n)$ converges but $\sum_{n=1}^\infty u_n$ diverges

asked May 12, 2023 at 7:53

9 votes

1 answer

413 views

$x_n=f\left(\frac{1}{n^2}\right)+f\left(\frac{2}{n^2}\right)+\cdots+f\left(\frac{n}{n^2}\right)$, $\lim_{n\to\infty}x_n=\dfrac{f'(0)}{2}$

asked Nov 16, 2021 at 0:16

4 votes

2 answers

130 views

Let $f(x)\in C^2[0,2]$, prove that there exists a $\xi\in(0,2)$ such that $\int_0^2f(x)dx=2f(1)+\frac{f''(\xi)}{3}$.

asked Dec 14, 2023 at 0:59

4 votes

1 answer

213 views

Tauberian Theorem and Prime Number Theorem

asked Jul 20, 2020 at 9:01

3 votes

2 answers

118 views

Is the function $f(x)=\sqrt[x+1]{\Gamma(x+2)}-\sqrt[x]{\Gamma(x+1)}$ decreasing?

asked Jul 9, 2021 at 4:08

3 votes

1 answer

145 views

$1^\alpha+2^\alpha+3^\alpha+\cdots+n^\alpha$

asked May 26 at 2:21

2 votes

0 answers

57 views

The function space $C^k[a,b]$ and $C^{k+1}[a,b]$

asked Sep 14, 2021 at 1:54

2 votes

1 answer

89 views

Continuous function of multivariable which has directional derivative nowhere

asked Mar 10, 2023 at 15:38

2 votes

0 answers

86 views

Great Picard's Theorem and infinite number of poles

asked Feb 8, 2021 at 2:10

2 votes

1 answer

583 views

A question about Riemann Integral, the function is not integrable, but the limit of Riemann sum exists

asked Apr 25, 2021 at 6:37

2 votes

1 answer

67 views

Let $U$ be a subset of $\mathbb{R}^2\setminus\{0\}$, $i_\ast(\pi_1(U))=0$ or $\mathbb{Z}$

asked Mar 24, 2020 at 16:50

2 votes

1 answer

257 views

Must a continuous function with finitely many strict extrema on a closed interval have bounded variation? And Pointwise convergence of Fourier series

asked Jun 19, 2021 at 13:59

1 vote

0 answers

76 views

Can we prove that $\int_0^1\int_0^1 \sin(mx)\sin(ny)|y-x|^{\alpha} \,dx\,dy$ is nonzero?

asked Sep 3, 2020 at 4:00

1 vote

1 answer

120 views

Uniform convergence about Dirichlet integral $f(s):=\int_1^\infty\frac{a(x)}{x^s}\,dx =\lim\limits_{T\to\infty}\int_1^T\frac{a(x)}{x^s}\,dx$

asked Mar 24, 2022 at 7:25

1 vote

1 answer

300 views

The necessary and sufficient conditions for a complex function to have primitive function

asked Apr 15, 2022 at 2:48

1 vote

0 answers

97 views

$\sum\limits_{n=1}^\infty\frac{a_n}{n}$ is convergent implies $\sum\limits_{n=1}^\infty\frac{a_n}{n^{1+it}}$ is convergent

asked Sep 25, 2022 at 6:07

1 vote

1 answer

293 views

On the conditions of the Implicit Function Theorem

asked Feb 23, 2023 at 1:11

1 vote

1 answer

44 views

Proving $\left[a_1,\frac{b_1+\sqrt\Delta}2\right]\cdot\left[a_2,\frac{b_2+\sqrt\Delta}2\right]=\left[a_1a_2,\frac{b+\sqrt\Delta}2\right]$

asked Aug 31 at 4:57

0 votes

1 answer

61 views

It a rotation matrix in 3D space a product of three basic rotations?

asked Mar 5 at 14:51

0 votes

2 answers

89 views

What is the determinant of Bernoulli matrix?

asked Jun 8, 2023 at 22:36

0 votes

1 answer

52 views

About Bessel's Book 1830: Tabulae Regiomontanae reductionum observationum astronomicarum ab anno 1750 usque ad annum 1850 computatae [closed]

asked Sep 3, 2023 at 12:57

0 votes

2 answers

128 views

If the second homotopy group is trivial, surface integral is independent with surface

asked Apr 18, 2023 at 0:43

0 votes

2 answers

91 views

Double Integrals in Polar Coordinates and the geometric meaning

asked Apr 12, 2021 at 15:38

0 votes

0 answers

42 views

A question about the regular condition of line integral

asked Apr 26, 2021 at 7:48

Stack Exchange Network

25 Questions

Find the limit $\lim\limits_{s\to0^+}\sum_{n=1}^\infty\frac{\sin n}{n^s}$

$\sum_{n=1}^\infty\ln(1+u_n)$ converges but $\sum_{n=1}^\infty u_n$ diverges

$x_n=f\left(\frac{1}{n^2}\right)+f\left(\frac{2}{n^2}\right)+\cdots+f\left(\frac{n}{n^2}\right)$, $\lim_{n\to\infty}x_n=\dfrac{f'(0)}{2}$

Let $f(x)\in C^2[0,2]$, prove that there exists a $\xi\in(0,2)$ such that $\int_0^2f(x)dx=2f(1)+\frac{f''(\xi)}{3}$.

Tauberian Theorem and Prime Number Theorem

Is the function $f(x)=\sqrt[x+1]{\Gamma(x+2)}-\sqrt[x]{\Gamma(x+1)}$ decreasing?

$1^\alpha+2^\alpha+3^\alpha+\cdots+n^\alpha$

The function space $C^k[a,b]$ and $C^{k+1}[a,b]$

Continuous function of multivariable which has directional derivative nowhere

Great Picard's Theorem and infinite number of poles

A question about Riemann Integral, the function is not integrable, but the limit of Riemann sum exists

Let $U$ be a subset of $\mathbb{R}^2\setminus\{0\}$, $i_\ast(\pi_1(U))=0$ or $\mathbb{Z}$

Must a continuous function with finitely many strict extrema on a closed interval have bounded variation? And Pointwise convergence of Fourier series

Can we prove that $\int_0^1\int_0^1 \sin(mx)\sin(ny)|y-x|^{\alpha} \,dx\,dy$ is nonzero?

Uniform convergence about Dirichlet integral $f(s):=\int_1^\infty\frac{a(x)}{x^s}\,dx =\lim\limits_{T\to\infty}\int_1^T\frac{a(x)}{x^s}\,dx$

The necessary and sufficient conditions for a complex function to have primitive function

$\sum\limits_{n=1}^\infty\frac{a_n}{n}$ is convergent implies $\sum\limits_{n=1}^\infty\frac{a_n}{n^{1+it}}$ is convergent

On the conditions of the Implicit Function Theorem

Proving $\left[a_1,\frac{b_1+\sqrt\Delta}2\right]\cdot\left[a_2,\frac{b_2+\sqrt\Delta}2\right]=\left[a_1a_2,\frac{b+\sqrt\Delta}2\right]$

It a rotation matrix in 3D space a product of three basic rotations?

What is the determinant of Bernoulli matrix?

About Bessel's Book 1830: Tabulae Regiomontanae reductionum observationum astronomicarum ab anno 1750 usque ad annum 1850 computatae [closed]

If the second homotopy group is trivial, surface integral is independent with surface

Double Integrals in Polar Coordinates and the geometric meaning

A question about the regular condition of line integral