E.H.E's user avatar
E.H.E's user avatar
E.H.E's user avatar
E.H.E
  • Member for 9 years, 4 months
  • Last seen more than a week ago
90 votes
4 answers
6k views

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

48 votes
1 answer
2k views

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

25 votes
2 answers
950 views

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

18 votes
3 answers
2k views

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

17 votes
3 answers
12k views

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

16 votes
1 answer
465 views

Closed-form of infinite continued fraction involving factorials

15 votes
4 answers
3k views

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

14 votes
1 answer
2k views

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

12 votes
2 answers
426 views

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

11 votes
3 answers
358 views

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

11 votes
3 answers
339 views

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

11 votes
4 answers
280 views

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

10 votes
1 answer
570 views

Can this interesting property be proven?

10 votes
4 answers
730 views

Which expansion of $e$ is more accurate?

10 votes
2 answers
522 views

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

10 votes
2 answers
226 views

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

10 votes
2 answers
437 views

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

10 votes
2 answers
291 views

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

9 votes
6 answers
529 views

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

9 votes
3 answers
288 views

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

8 votes
2 answers
208 views

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

8 votes
3 answers
213 views

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

8 votes
3 answers
612 views

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

8 votes
3 answers
542 views

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

8 votes
1 answer
168 views

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

8 votes
3 answers
339 views

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

7 votes
1 answer
189 views

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

7 votes
4 answers
984 views

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

7 votes
3 answers
436 views

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

7 votes
1 answer
149 views

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

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