User Jacob - Mathematics Stack Exchange

8 votes

Accepted

Show that $\int_{-\infty}^{\infty} \frac{\sin(2\pi(x-a))}{\pi(x-a)}\frac{\sin(2\pi(x-b))}{\pi(x-b)}dx = \frac{\sin2\pi(a-b)}{\pi(a-b)}$

answered Jan 20, 2022 at 23:44

7 votes

Twin prime conjecture proof error

answered May 16, 2016 at 16:53

7 votes

On the summation $\sum \limits_{n=1}^{\infty} \arctan \left ( \frac{1}{n^3+n^2+n+1} \right )$

answered Oct 29, 2017 at 23:33

6 votes

Weird property of quadratic equations

answered Apr 6, 2016 at 1:33

6 votes

Accepted

Closed form for $\frac{x}{1+x} +\frac{x^2}{1+x+x^2} +\frac{x^3}{1+x+x^2+x^3}+...$

answered Apr 20, 2018 at 18:02

5 votes

Is each of $\int_0^\infty\frac{dx}{x^x},\int_0^\infty\frac{dx}{x^{x^{x^x}}},\int_0^\infty\frac{dx}{x^{x^{x^{x^{x^x}}}}},\cdots$ less than $2$?

answered Feb 22, 2022 at 4:17

5 votes

Accepted

Average value of $|x_1|$ over $n$-sphere

answered Aug 28, 2022 at 17:56

5 votes

Accepted

One-point compactification of $S^3\setminus S^1$

answered Oct 19, 2022 at 7:33

4 votes

Accepted

Triple product $\displaystyle\prod_{k=1}^{\infty} \prod_{n=1}^{\infty} \prod_{m=1}^{\infty}(k+n+m)^{\frac{(-1)^{k+m+n}}{k+m+n}}$

answered Nov 4, 2021 at 18:27

4 votes

Accepted

How to analyze $(-1)^{\left \lfloor n\theta \right \rfloor}$ (in which $\theta$ is an irrational number)?

answered Nov 10, 2018 at 4:13

4 votes

Evaluate $\lim_{n\to \infty}(0.9999+\frac{1}{n})^n$

answered Nov 8, 2018 at 22:10

4 votes

Accepted

Analytic continuation of partitions generating function

answered Nov 14, 2018 at 1:39

3 votes

Accepted

Is $\phi(ab)\ge\phi(a)\cdot \phi(b)$ true for every positive integer pair $(a/b)$?

answered Oct 25, 2017 at 12:44

3 votes

The sum of $9$ shapes is $30$. There are $6$ circles and $3$ squares. What are the values of the shapes?

answered Mar 26, 2017 at 0:02

3 votes

Infinite product $\prod \frac{R^{2^n}+z^{z^n}}{R^{2^n}}$ coverges

answered Feb 26, 2018 at 18:18

3 votes

How to find the sum of the series $(1) + (2 + 3) +(4 + 5 + 6) + \cdots$ to $n$ terms?

answered May 7, 2017 at 18:59

3 votes

Accepted

Asymptotic formula for $\sum_{k=1}^n \frac{1}{\varphi(k)}$?

answered Jan 28, 2019 at 18:20

3 votes

Picking points on a sphere at random

answered Nov 2, 2021 at 7:06

2 votes

Accepted

Little "oh" and prime number theorem

answered Oct 25, 2021 at 6:04

2 votes

Area under the function $f$

answered Nov 3, 2021 at 15:34

2 votes

Intuition behind $ ||x|-|y|| \leq |x-y| $

answered Feb 13, 2019 at 22:36

2 votes

Prove $\sum_{d | n} \mu(d) (\log(d))^2=0$

answered Oct 20, 2021 at 4:03

2 votes

Accepted

How to separate $ \prod_{k=1}^{\infty} \frac{(16k-15)^{\frac{1}{16k-15}}}{(16k-1)^{\frac{1}{16k-1}}} $ from Log-gamma function.

answered Nov 14, 2021 at 22:24

2 votes

Accepted

A set of points is contained in a sphere $S$. When is $S$ also the circumsphere?

answered Dec 14, 2021 at 18:33

2 votes

Accepted

Bijection between $\{0,1\}^\omega$ and the set of positive integers

answered Dec 20, 2021 at 20:15

2 votes

Proving $ \sum_{n=-\infty}^{\infty}(-1)^n e^{-\left(x - \frac{n}{2}\right)^2} = K\cos(2\pi x)$

answered Apr 14, 2018 at 19:23

2 votes

Evaluate the infinite product $\prod_{k \geq 2}\sqrt[k]{1+\frac{1}{k}}=\sqrt{1+\frac{1}{2}} \sqrt[3]{1+\frac{1}{3}} \sqrt[4]{1+\frac{1}{4}} \cdots$

answered May 11, 2016 at 20:27

2 votes

Accepted

Prove this series equal to this integral $\sum_{n=0}^\infty{\frac{(-1)^ny^{nx+1}}{nx+1}} = \int_{0}^y{\frac{dt}{1+t^x}}$

answered Apr 15, 2018 at 2:34

2 votes

Accepted

Show that $\lim_{n\to\infty}\mathcal I\left(\exp\left(\frac{2\pi\cdot i}{\log_n(p_n\#)}\right)\right)=0$

answered Sep 2, 2018 at 22:08

2 votes

Accepted

Solving $x^{-x^{1-x}}=\sqrt[\sqrt2]{2}$

answered Feb 25, 2022 at 12:39

Stack Exchange Network

49 Answers

Show that $\int_{-\infty}^{\infty} \frac{\sin(2\pi(x-a))}{\pi(x-a)}\frac{\sin(2\pi(x-b))}{\pi(x-b)}dx = \frac{\sin2\pi(a-b)}{\pi(a-b)}$

Twin prime conjecture proof error

On the summation $\sum \limits_{n=1}^{\infty} \arctan \left ( \frac{1}{n^3+n^2+n+1} \right )$

Weird property of quadratic equations

Closed form for $\frac{x}{1+x} +\frac{x^2}{1+x+x^2} +\frac{x^3}{1+x+x^2+x^3}+...$

Is each of $\int_0^\infty\frac{dx}{x^x},\int_0^\infty\frac{dx}{x^{x^{x^x}}},\int_0^\infty\frac{dx}{x^{x^{x^{x^{x^x}}}}},\cdots$ less than $2$?

Average value of $|x_1|$ over $n$-sphere

One-point compactification of $S^3\setminus S^1$

Triple product $\displaystyle\prod_{k=1}^{\infty} \prod_{n=1}^{\infty} \prod_{m=1}^{\infty}(k+n+m)^{\frac{(-1)^{k+m+n}}{k+m+n}}$

How to analyze $(-1)^{\left \lfloor n\theta \right \rfloor}$ (in which $\theta$ is an irrational number)?

Evaluate $\lim_{n\to \infty}(0.9999+\frac{1}{n})^n$

Analytic continuation of partitions generating function

Is $\phi(ab)\ge\phi(a)\cdot \phi(b)$ true for every positive integer pair $(a/b)$?

The sum of $9$ shapes is $30$. There are $6$ circles and $3$ squares. What are the values of the shapes?

Infinite product $\prod \frac{R^{2^n}+z^{z^n}}{R^{2^n}}$ coverges

How to find the sum of the series $(1) + (2 + 3) +(4 + 5 + 6) + \cdots$ to $n$ terms?

Asymptotic formula for $\sum_{k=1}^n \frac{1}{\varphi(k)}$?

Picking points on a sphere at random

Little "oh" and prime number theorem

Area under the function $f$

Intuition behind $ ||x|-|y|| \leq |x-y| $

Prove $\sum_{d | n} \mu(d) (\log(d))^2=0$

How to separate $ \prod_{k=1}^{\infty} \frac{(16k-15)^{\frac{1}{16k-15}}}{(16k-1)^{\frac{1}{16k-1}}} $ from Log-gamma function.

A set of points is contained in a sphere $S$. When is $S$ also the circumsphere?

Bijection between $\{0,1\}^\omega$ and the set of positive integers

Proving $ \sum_{n=-\infty}^{\infty}(-1)^n e^{-\left(x - \frac{n}{2}\right)^2} = K\cos(2\pi x)$

Evaluate the infinite product $\prod_{k \geq 2}\sqrt[k]{1+\frac{1}{k}}=\sqrt{1+\frac{1}{2}} \sqrt[3]{1+\frac{1}{3}} \sqrt[4]{1+\frac{1}{4}} \cdots$

Prove this series equal to this integral $\sum_{n=0}^\infty{\frac{(-1)^ny^{nx+1}}{nx+1}} = \int_{0}^y{\frac{dt}{1+t^x}}$

Show that $\lim_{n\to\infty}\mathcal I\left(\exp\left(\frac{2\pi\cdot i}{\log_n(p_n\#)}\right)\right)=0$

Solving $x^{-x^{1-x}}=\sqrt[\sqrt2]{2}$