User E.H.E - Mathematics Stack Exchange

92 votes

4 answers

6k views

How to prove that $\frac{\zeta(2) }{2}+\frac{\zeta (4)}{2^3}+\frac{\zeta (6)}{2^5}+\frac{\zeta (8)}{2^7}+\cdots=1$?

asked Dec 24, 2014 at 18:24

48 votes

1 answer

2k views

Evaluate $ \int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}dx $

asked Nov 2, 2014 at 7:47

25 votes

2 answers

953 views

Limit of $\lim_{x\rightarrow 1}\sqrt{x-\sqrt[3]{x-\sqrt[4]{x-\sqrt[5]{x-.....}}}}$

asked Nov 30, 2014 at 17:58

18 votes

3 answers

2k views

How can I prove $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}...}}}}=2$ [duplicate]

asked Dec 22, 2014 at 13:06

17 votes

3 answers

12k views

Finding the all roots of a polynomial by using Newton-Raphson method.

asked Oct 30, 2014 at 14:56

16 votes

1 answer

465 views

Closed-form of infinite continued fraction involving factorials

asked Dec 19, 2014 at 11:07

16 votes

5 answers

3k views

Proving $\sum_{n=1}^{\infty }\frac{\cos(n)}{n^4}=\frac{\pi ^4}{90}-\frac{\pi ^2}{12}+\frac{\pi }{12}-\frac{1}{48}$

asked Mar 28, 2015 at 22:11

14 votes

1 answer

2k views

Prove without using a calculator $(\ln 6)^{(\ln 5)^{(\ln 4)^{(\ln 3)^{(\ln 2)}}}}<\pi$

asked Oct 23, 2015 at 11:18

11 votes

2 answers

434 views

Is there a closed-form of $\frac{\zeta (2)}{\pi ^2}+\frac{\zeta (4)}{\pi ^4}+\frac{\zeta (6)}{\pi ^6}+.....$

asked Sep 14, 2015 at 14:01

11 votes

3 answers

358 views

How to prove $(\frac{1}{5^3}-\frac{1}{7^3})+(\frac{1}{11^3}-\frac{1}{13^3})+(\frac{1}{17^3}-\frac{1}{19^3})+...=(1-\frac{\pi ^3}{18\sqrt{3}})$

asked Jan 14, 2015 at 18:51

11 votes

3 answers

343 views

How can prove that $\sum_{n=1}^{\infty }\frac{\zeta (2n)}{4^{n-1}}(1-\frac{1}{4^n})=\frac{\pi }{2}$

asked Dec 26, 2014 at 13:10

11 votes

4 answers

285 views

Proving $1+2^n+3^n+4^n$ is divisible by $10$

asked Dec 10, 2014 at 12:02

10 votes

1 answer

571 views

Can this interesting property be proven?

asked Dec 15, 2014 at 19:13

10 votes

4 answers

777 views

Which expansion of $e$ is more accurate?

asked Nov 29, 2014 at 13:03

10 votes

2 answers

525 views

Proving $\left(\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}}\right)\left(\sqrt{x-\sqrt{x-\sqrt{x-\sqrt{x+\cdots}}}}\right)=x$

asked Dec 4, 2014 at 17:33

10 votes

2 answers

228 views

How can I prove analytically the number $2^{100000}+1$ is not prime??

asked Dec 5, 2014 at 23:33

10 votes

2 answers

439 views

Proving that $\frac{\pi ^2}{p}\neq \sum_{n=1}^{\infty }\frac{1}{a_{n}^2}$

asked Nov 6, 2014 at 9:35

10 votes

2 answers

296 views

Proving $\zeta(2) - \beta(1) + \zeta(4) - \beta(3) + \zeta(6)- \beta(5) + \ldots=1$

asked Feb 22, 2015 at 19:40

9 votes

6 answers

541 views

Finding the positive integer numbers to get $\frac{\pi ^2}{9}$

asked Nov 5, 2014 at 14:30

9 votes

3 answers

288 views

Proving this formula $1+\sum_{n=0}^{\infty }\frac{1}{\pi \left(2n+\frac{3}{4}\right)\left(2n+\frac{5}{4}\right)}=\sqrt2$

asked Nov 3, 2014 at 8:04

8 votes

2 answers

208 views

Can help me to find $\sum_{n=1}^{\infty }\frac{1}{(4n-1)^3}$?

asked Oct 28, 2014 at 1:01

8 votes

3 answers

216 views

How can I prove $\sqrt{(111...)+(55...)^2}=5...6$

asked Dec 3, 2014 at 13:52

8 votes

3 answers

627 views

Solution to differential equations $y(0)=1$ and $y^{(n)}=y+1$

asked Dec 14, 2014 at 12:45

8 votes

3 answers

621 views

Show that $\pi =4-\sum_{n=1}^{\infty }\frac{(n)!(n-1)!}{(2n+1)!}2^{n+1}$

asked Mar 23, 2015 at 21:42

8 votes

1 answer

172 views

Proving that $\sum_{n=0}^{\infty }\frac{3(n!)^2}{(2n+2)!}=\sum_{n=1}^{\infty }\frac{1}{n^2}=\frac{\pi ^2}{6}$

asked Sep 28, 2015 at 22:57

8 votes

3 answers

339 views

Is $\sum_{n=1}^{\infty }\frac{8n\cdot\zeta (2n)}{3\cdot 2^{2n}}=\zeta (2)$?

asked Sep 29, 2015 at 18:35

7 votes

1 answer

189 views

Proving that $(1+10^n)$cannot be a prime number when $(n>2)$

asked Sep 23, 2015 at 11:38

7 votes

4 answers

1k views

What is the $\lim_{n\rightarrow \infty }(1+\frac{1}{n})^{n^n}$

asked May 22, 2015 at 22:53

7 votes

3 answers

454 views

Proving $\sum_{n =1,3,5..}^{\infty }\frac{4k \ \sin^2\left(\frac{n}{k}\right)}{n^2}=\pi$

asked Feb 17, 2015 at 15:02

7 votes

1 answer

149 views

Showing $\sum_{n=1}^{\infty }(-1)^{n-1}\frac{(2n-2)!\zeta (2n)}{\pi ^{2n}}(1-\frac{1}{2^{2n}})(1+\frac{1}{2^{2n-1}})=\frac{\log(2)}{4}$.

asked Mar 9, 2015 at 13:05

Stack Exchange Network

136 Questions

How to prove that $\frac{\zeta(2) }{2}+\frac{\zeta (4)}{2^3}+\frac{\zeta (6)}{2^5}+\frac{\zeta (8)}{2^7}+\cdots=1$?

Evaluate $ \int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}dx $

Limit of $\lim_{x\rightarrow 1}\sqrt{x-\sqrt[3]{x-\sqrt[4]{x-\sqrt[5]{x-.....}}}}$

How can I prove $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}...}}}}=2$ [duplicate]

Finding the all roots of a polynomial by using Newton-Raphson method.

Closed-form of infinite continued fraction involving factorials

Proving $\sum_{n=1}^{\infty }\frac{\cos(n)}{n^4}=\frac{\pi ^4}{90}-\frac{\pi ^2}{12}+\frac{\pi }{12}-\frac{1}{48}$

Prove without using a calculator $(\ln 6)^{(\ln 5)^{(\ln 4)^{(\ln 3)^{(\ln 2)}}}}<\pi$

Is there a closed-form of $\frac{\zeta (2)}{\pi ^2}+\frac{\zeta (4)}{\pi ^4}+\frac{\zeta (6)}{\pi ^6}+.....$

How to prove $(\frac{1}{5^3}-\frac{1}{7^3})+(\frac{1}{11^3}-\frac{1}{13^3})+(\frac{1}{17^3}-\frac{1}{19^3})+...=(1-\frac{\pi ^3}{18\sqrt{3}})$

How can prove that $\sum_{n=1}^{\infty }\frac{\zeta (2n)}{4^{n-1}}(1-\frac{1}{4^n})=\frac{\pi }{2}$

Proving $1+2^n+3^n+4^n$ is divisible by $10$

Can this interesting property be proven?

Which expansion of $e$ is more accurate?

Proving $\left(\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}}\right)\left(\sqrt{x-\sqrt{x-\sqrt{x-\sqrt{x+\cdots}}}}\right)=x$

How can I prove analytically the number $2^{100000}+1$ is not prime??

Proving that $\frac{\pi ^2}{p}\neq \sum_{n=1}^{\infty }\frac{1}{a_{n}^2}$

Proving $\zeta(2) - \beta(1) + \zeta(4) - \beta(3) + \zeta(6)- \beta(5) + \ldots=1$

Finding the positive integer numbers to get $\frac{\pi ^2}{9}$

Proving this formula $1+\sum_{n=0}^{\infty }\frac{1}{\pi \left(2n+\frac{3}{4}\right)\left(2n+\frac{5}{4}\right)}=\sqrt2$

Can help me to find $\sum_{n=1}^{\infty }\frac{1}{(4n-1)^3}$?

How can I prove $\sqrt{(111...)+(55...)^2}=5...6$

Solution to differential equations $y(0)=1$ and $y^{(n)}=y+1$

Show that $\pi =4-\sum_{n=1}^{\infty }\frac{(n)!(n-1)!}{(2n+1)!}2^{n+1}$

Proving that $\sum_{n=0}^{\infty }\frac{3(n!)^2}{(2n+2)!}=\sum_{n=1}^{\infty }\frac{1}{n^2}=\frac{\pi ^2}{6}$

Is $\sum_{n=1}^{\infty }\frac{8n\cdot\zeta (2n)}{3\cdot 2^{2n}}=\zeta (2)$?

Proving that $(1+10^n)$cannot be a prime number when $(n>2)$

What is the $\lim_{n\rightarrow \infty }(1+\frac{1}{n})^{n^n}$

Proving $\sum_{n =1,3,5..}^{\infty }\frac{4k \ \sin^2\left(\frac{n}{k}\right)}{n^2}=\pi$

Showing $\sum_{n=1}^{\infty }(-1)^{n-1}\frac{(2n-2)!\zeta (2n)}{\pi ^{2n}}(1-\frac{1}{2^{2n}})(1+\frac{1}{2^{2n-1}})=\frac{\log(2)}{4}$.