User:Zubair12345

user:zubair1234

user:The-problem

Definitions
$$\phi=\frac{\sqrt{5}+1}{2}$$

A is the Glaisher - Kinkelin Constant 

ψ is the triggamma function

K is the Catalan's constant

B2n is Bernoulli numbers : B2 = 1\6, B4 = 1\30 , B6 = 1\42 , ...

$$\Psi(q)=\sum_{n=0}^{\infty}q^{\frac{n(n+1)}{2}}$$

Triangular and Tetrahedral numbers
Tn = 1, 3 , 6 , 10 , 15 , ....

Te(k) = 1, 4 , 10 , 20 , 35 , ....

$$\sum_{r=1}^{M}(n+r)=T_{n+M}-T_{n}$$

$$\sum_{r=1}^{k}(n+r)^2=k(T_n+T_{n+k})-Te_{k-1}$$

$$\sum_{r=1}^{M}(n+r)^3=T_{n+M}^2-T_{n}^2$$

Others

$$\sum_{k=1}^{n}{[pk-(p-1)]}=n^2+(p-2)\cdot{T_{n-1}}$$

Special cases

$$\sum_{k=1}^{n}{(2k-1)}=n^2$$

$$\sum_{k=1}^{n}{k}=n^2-T_{n-1}$$

$$\sum_{k=1}^{n}{(2k-1)^2}=Te_{2n-1}$$

$$\sum_{k=1}^{n}{(2k)^2}=Te_{2n}$$

Binomial coefficients

$$1\cdot{T}_n+2\cdot{T_{n+1}}+1\cdot{T_{n+2}}=(n+1)^2+(n+2)^2$$

Others

$$T_nT_{n+1}+T_nT_{n+2}+T_{n+1}T_{n+2}=3T_{n+1}^2$$

$$T_n+T_{n+1}^2+T_{n+2}=\frac{1}{4}(n^2+3n+4)^2$$

$$1^3+3^3+6^2=4^3$$

$$3^3+6^3+10^2=7^3$$

Constant of continued fraction
$$\frac{\sum_{k=0}^{\infty}\frac{x+k}{(k!)^2}}{ \sum_{k=0}^{\infty}\frac{1}{(k!)^2} }= x+\frac{1}{1+\frac{1}{2+\frac{1}{3+\frac{1}{4+\cdots}}}}$$

Factorial product
Triangular numbers

$$T_k=k+\left(k-1\right)+\cdots{+}2+1$$

N ≥ 3

$$2^{N+3}\prod_{k=1}^{N}T_{2k}=(2N+2)!$$

Example : N = 3 

$$2^{6}\prod_{k=1}^{3}T_{2k}=(8)!$$

$$2^{6}\cdot{T_{2}}\cdot{T_{4}}\cdot{T_{6}}=(8)!$$

$$2^{6}\cdot{3}\cdot{10}\cdot{21}=(8)!$$

Other case

$$2^2\prod_{k=1}^{4}T_k=6!$$

$$2^6\prod_{k=1}^{6}T_k=10!$$

$$2^N\prod_{k=1}^{N}T_k=N!(N+1)!$$

$$\left(1!\cdot{3!}\right)^{k+2}\cdot{4}\cdot{5}=6!(3!)^k$$

$$\left(1!\cdot{3!}\cdot{5!}\right)^{k+2}\cdot{7}=10!(6!)^k$$

1
$$1=\sqrt{3}-\sqrt{\frac{2n}{3n+n\sqrt{\frac{2n}{3n+\cdots}}}}$$

$$1=\phi^2+\sqrt{\frac{\phi}{\phi^2+\sqrt{\frac{\phi} {\phi^2+\sqrt{\frac{\phi}{\phi^2+\cdots}}}}}}- \sqrt{\phi+\phi\sqrt{\phi+\phi\sqrt{\phi+\cdots}}}$$

$$1=\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}- \sqrt{\frac{1}{2+\sqrt{\frac{1} {2+\sqrt{\frac{1}{2+\cdots}}}}}}$$

In general

$$k=\sqrt{k+\sqrt{k+\sqrt{k+\cdots}}}- \sqrt{\frac{k}{k+1+\sqrt{\frac{k} {k+1+\sqrt{\frac{k}{k+1+\cdots}}}}}}$$

$$1=\sqrt{\frac{\phi^{k+1}}{\phi^{k+2}-\phi^k\sqrt{\frac{\phi^{k+1}} {\phi^{k+2}-\phi^k\sqrt{\frac{\phi^{k+1}}{\phi^{k+2}-\cdots}}}}}}$$

$$1=\sqrt{\frac{\phi}{\phi^2-\sqrt{\frac{\phi} {\phi^2-\sqrt{\frac{\phi}{\phi^2-\cdots}}}}}}$$

Nth Radical
'''Fibonacci sequence : 1, 1 , 2 , 3 ,. . .'''

F1 = 1, F2 = 1

$$\phi=\sqrt[k]{\phi{F_{k-2}}+\phi{F_{k-1}}\sqrt[k]{\phi{F_{k-2}}+\phi{F_{k-1}}\sqrt[k] {\phi{F_{k-2}}+\cdots}}}$$

$$2=\sqrt[3]{2+3\sqrt[3]{2+3\sqrt[3]{2+3\sqrt[3]{2+\cdots}}}}$$

General

$$n=\sqrt[k]{mn+(n^{k-1}-m)\sqrt[k]{mn+(n^{k-1}-m)\sqrt[k]{mn+\cdots}}}$$

Integer Continued fraction
$$k+1=k+\frac{k+2}{k+1+\frac{k+3}{k+2+\frac{k+4}{k+3+\cdots}}}$$

$$1=\frac{2}{1+\frac{3}{2+\frac{4}{3+\cdots}}}$$

$$k+1+\frac{1}{k+3}=k+\frac{k+3}{k+1+\frac{k+4}{k+2+\frac{k+5}{k+3+\cdots}}}$$

$$2=\frac{1}{-1+\frac{2}{0+\frac{3}{1+\cdots}}}$$

$$k+1+\frac{2k+8}{k^2+7k+13}=k+\frac{k+4}{k+1+\frac{k+5}{k+2+\frac{k+6}{k+3+\cdots}}}$$

$$2=\frac{3-\frac{1}{\phi^2}}{0-\frac{1}{\phi^2}+\frac{4-\frac{1}{\phi^2}} {1-\frac{1}{\phi^2}+\frac{5-\frac{1}{\phi^2}}{2-\frac{1}{\phi^2}+\cdots}}}$$

$$3=\frac{1}{-2+\frac{2}{-1+\frac{3}{0+\frac{4}{1+\frac{5}{2+\frac{6}{3+\cdots}}}}}}$$

General: Integer

$$n=\frac{1}{1-n+\frac{2}{2-n+\frac{3}{3-n+\frac{4}{4-n+\frac{5}{5-n+\cdots}}}}}$$

$$\sqrt{n+1}=\frac{n-\frac{1}{\phi^k}}{0-\frac{1}{\phi^k}+\frac{n+1-\frac{1}{\phi^k}} {1-\frac{1}{\phi^k}+\frac{n+2-\frac{1}{\phi^k}}{2-\frac{1}{\phi^k}+\cdots}}}$$

$$1=\frac{1-\frac{1}{\phi}}{0-\frac{1}{\phi}+\frac{2-\frac{1}{\phi}} {1-\frac{1}{\phi}+\frac{3-\frac{1}{\phi}}{2-\frac{1}{\phi}+\cdots}}}$$

$$1=\frac{1+\frac{1}{\phi^2}}{0+\frac{1}{\phi^2}+\frac{2+\frac{1}{\phi^2}} {1+\frac{1}{\phi^2}+\frac{3+\frac{1}{\phi^2}}{2+\frac{1}{\phi^2}+\cdots}}}$$

Others

$$\phi^2+\sqrt{\phi^4-1}=\phi^3+\sqrt{\frac{2\phi}{\phi^3+\sqrt{\frac{2\phi} {\phi^3+\sqrt{\frac{2\phi}{\phi^3+\cdots}}}}}}$$

$$2+\sqrt{3}=3+\sqrt{\frac{2}{3+\sqrt{\frac{2}{3+\sqrt{\frac{2}{3+\cdots}}}}}}$$

$$\phi^2=2+\sqrt{\frac{1}{2+\sqrt{\frac{1}{2+\sqrt{\frac{1}{2+\cdots}}}}}}$$

$$\phi^4=6+\sqrt{\frac{5}{6+\sqrt{\frac{5}{6+\sqrt{\frac{5}{6+\cdots}}}}}}$$

In general

Lucas sequence : 2, 1 , 3 , 4 , 7 , ....

L1 = 2, L2 = 1 

$$\phi^{2n}=L_{2n+1}-1+\sqrt{\frac{L_{2n+1}-2}{L_{2n+1}-1+\sqrt{\frac{L_{2n+1}-2} {L_{2n+1}-1+\sqrt{\frac{L_{2n+1}-2}{L_{2n+1}-1+\cdots}}}}}}$$

In general

$$n+1+\sqrt{n(n+2)}=2n+1+\sqrt{\frac{2n}{2n+1+\sqrt{\frac{2n}{2n+1+ \sqrt{\frac{2n}{2n+1+\cdots}}}}}}$$

Nested Series and its continued fraction
$$0!^nk-[0^n[0]+1]=1+\frac{1}{2^n}\left(1+\frac{1}{3^n}\left(1+\frac{1}{4^n} \left(1+\cdots\right)\right)\right)$$

$$1!^nk-[1^n[0^n[0]+1]+1]=\frac{1}{2^n}+\frac{1}{3^n}\left(\frac{1}{2^n}+\frac{1}{4^n} \left(\frac{1}{2^n}+\frac{1}{4^n} \left(\frac{1}{2^n}+\cdots\right)\right)\right)$$

$$2!^nk-[2^n[1^n[0^n[0]+1]+1]+1]=\frac{1}{3^n}+\frac{1}{4^n}\left(\frac{1}{3^n}+\frac{1}{5^n} \left(\frac{1}{3^n}+\frac{1}{6^n} \left(\frac{1}{3^n}+\cdots\right)\right)\right)$$

And so on ...

special condition n = 1

$$0!k-1=1+\frac{1}{2}\left(1+\frac{1}{3}\left(1+\frac{1}{4} \left(1+\cdots\right)\right)\right)$$

$$1!k-2=\frac{1}{2}+\frac{1}{3}\left(\frac{1}{2}+\frac{1}{4} \left(\frac{1}{2}+\frac{1}{4} \left(\frac{1}{2}+\cdots\right)\right)\right)$$

$$2!k-5=\frac{1}{3}+\frac{1}{4}\left(\frac{1}{3}+\frac{1}{5} \left(\frac{1}{3}+\frac{1}{6} \left(\frac{1}{3}+\cdots\right)\right)\right)$$

It is known this is an exponential series

$$0!e-1=1+\frac{1}{2}\left(1+\frac{1}{3}\left(1+\frac{1}{4} \left(1+\cdots\right)\right)\right)$$

$$1!e-2=\frac{1}{2}+\frac{1}{3}\left(\frac{1}{2}+\frac{1}{4} \left(\frac{1}{2}+\frac{1}{4} \left(\frac{1}{2}+\cdots\right)\right)\right)$$

$$2!e-5=\frac{1}{3}+\frac{1}{4}\left(\frac{1}{3}+\frac{1}{5} \left(\frac{1}{3}+\frac{1}{6} \left(\frac{1}{3}+\cdots\right)\right)\right)$$

Its continued fraction

The first equation is Ramanujan equation

$$\frac{1}{0!e-1}=0+\frac{1}{1+\frac{2}{2+\frac{3}{3+\frac{4}{4+\cdots}}}}$$

$$\frac{1}{1!e-2}=1+\frac{1}{2+\frac{2}{3+\frac{3}{4+\frac{4}{5+\cdots}}}}$$

$$\frac{1}{2!e-5}=2+\frac{1}{3+\frac{2}{4+\frac{3}{5+\frac{4}{6+\cdots}}}}$$

And so on ...

Variation of the continued fraction

$$\frac{2}{e-2}=2+\frac{3}{3+\frac{4}{4+\frac{5}{5+\frac{6}{6+\cdots}}}}$$

Approximation
'''Fib : 1 1 2 3 5 8 13 21 55 89, ... '''

Triangle number nth : 1 3 6 10 15 21, ...

$$\frac{\pi^2}{6}\approx{\frac{1}{F_1}+\frac{1}{F_3}+\frac{1}{F_6}+\frac{1}{F_{10}}+\cdots +\frac{1}{\left(239+\frac{1}{\phi^6}\right)^2}}$$

$$\frac{\pi^2}{6}\approx{\frac{1}{1}+\frac{1}{2}+\frac{1}{8}+\frac{1}{55}+\cdots +\frac{1}{\left(239+\frac{1}{\phi^6}\right)^2}}$$

'''Lucas : 2 1 3 4 7 11 18 29 47 123, ... '''

power of 2 nth : 1 2 4 8 16 32, ...

$$\frac{3}{2}+\frac{1}{2\sqrt{\frac{\sqrt{3}-1}{10}+3}} \approx{\frac{1}{L_1}+\frac{1}{L_2}+\frac{1}{L_4}+\frac{1}{L_{8}}+\cdots}$$

$$\frac{3}{2}+\frac{1}{2\sqrt{\frac{\sqrt{3}-1}{10}+3}} \approx{\frac{1}{2}+\frac{1}{1}+\frac{1}{4}+\frac{1}{29}+\cdots}$$

Nth root of phi
Fib : 1 1 2 3 5 8, ...

'''F_n : 1 2 3 4 5 6 ,... n'''

$$\phi=\sqrt[n]{F_{n-1}+F_{n}\sqrt[n]{F_{n-1}+F_{n}\sqrt[n]{F_{n-1}+F_{n} \sqrt[n]{F_{n-1}+\cdots}}}}$$

$$\phi=\sqrt[2]{1+1\sqrt[2]{1+1\sqrt[2]{1+1\sqrt[2]{1+\cdots}}}}$$

$$\phi=\sqrt[3]{1+2\sqrt[3]{1+2\sqrt[3]{1+2\sqrt[3]{1+\cdots}}}}$$

$$\phi=\sqrt[4]{2+3\sqrt[4]{2+3\sqrt[4]{2+3\sqrt[4]{2+\cdots}}}}$$

Prime product
'''Prime : 2 3 5 7 11 13 ,... Pk'''

'''nth  : 1 2 3 4 5 6 ,... k'''

$$\prod_{k=1}^{\infty}\frac{P^2_{2k-1}+1}{P^2_{2k-1}-1}=\frac{2^4-1}{2^3}$$

$$\prod_{k=1}^{\infty}\frac{P^2_{2k}+1}{P^2_{2k}-1}=\frac{2^2}{2^2-1}$$

Combine them Yields Ramanujan equation:

$$\prod_{k=1}^{\infty}\frac{P^2_k+1}{P^2_k-1}=\frac{2^2+1}{2}$$

Power of 2
$$p^{\frac{2n-1}{2}}+\frac{k}{2^m}=\sqrt{p^{2n-1}+\frac{k}{2^{m-1}}\sqrt{p^{2n-1}+ \frac{k}{2^{m+0}}\sqrt{p^{2n-1}+\frac{k}{2^{m+1}}\sqrt{p^{2n-1}+ \frac{k}{2^{m+2}}\sqrt{p^{2n-1}+\cdots}}}}}$$

$$p^n+\frac{k}{2^m}=\sqrt{p^{2n}+\frac{k}{2^{m-1}} \sqrt{p^{2n}+\frac{k}{2^{m+0}}\sqrt{p^{2n}+\frac{k}{2^{m+1}}\sqrt{p^{2n}+ \frac{k}{2^{m+2}}\sqrt{p^{2n}+\cdots}}}}}$$

$$2^{\frac{2n-1}{2}}+\frac{k}{2}=\sqrt{2^{2n-1}+k\sqrt{2^{2n-1}+ \frac{k}{2}\sqrt{2^{2n-1}+\frac{k}{4}\sqrt{2^{2n-1}+ \frac{k}{8}\sqrt{2^{2n-1}+\cdots}}}}}$$

$$2^n+\frac{k}{2}=\sqrt{2^{2n}+k\sqrt{2^{2n}+\frac{k}{2}\sqrt{2^{2n}+\frac{k}{4}\sqrt{2^{2n}+ \frac{k}{8}\sqrt{2^{2n}+\cdots}}}}}$$

n = 0, k = -1 Yields Ramanujan equation:

$$\frac{1}{2}=\sqrt{1-\sqrt{1-\frac{1}{2}\sqrt{1-\frac{1}{4}\sqrt{1-\frac{1}{8} \sqrt{1-\cdots}}}}}$$

Algorithms
A_n initial guess (any value)

$$A_{n+1}=\frac{A_n}{5}+\sqrt{5}=\frac{25}{8\phi-4}$$

$$A_{n+1}=\frac{A_n}{G}+\sqrt[m]{G}=\frac{G^2}{G-1}\cdot{\frac{1}{\sqrt[m]{G^{m-1}}}}$$

Infinite product of sin and cos
$$\sin\left(\frac{\pi}{2n+1}\right)\prod_{k=2}^{n}\sin\left(\frac{k\pi}{2n+1}\right)= \frac{\sqrt{2n+1}}{2^n}$$

$$\cos\left(\frac{\pi}{2n+1}\right)\prod_{k=2}^{n}\cos\left(\frac{k\pi}{2n+1}\right)= \frac{1}{2^n}$$

$$\tan\left(\frac{\pi}{2n+1}\right)\prod_{k=2}^{n}\tan\left(\frac{k\pi}{2n+1}\right)= \sqrt{2n+1}$$

$$\prod_{k=1}^{n}\sin\left(\frac{k\pi}{2n+2}\right)= \frac{\sqrt{n+1}}{2^n}$$

$$\prod_{k=1}^{n}\cos\left(\frac{k\pi}{2n+2}\right)= \frac{\sqrt{n+1}}{2^n}$$

$$\prod_{k=1}^{n}\tan\left(\frac{k\pi}{2n+2}\right)=1$$

Certain value
$$\prod_{k=1}^{3}\sin\left(\frac{k\pi}{4}\right)=\frac{1+\sqrt{1}}{2^2}$$

$$\prod_{k=1}^{5}\sin\left(\frac{k\pi}{8}\right)=\frac{\sqrt{2+\sqrt{2}}}{2^3}$$

$$\prod_{k=1}^{7}\sin\left(\frac{k\pi}{12}\right)=\frac{3+\sqrt{3}}{2^6}$$

$$\cos\left(\frac{2\pi}{7}\right)+\cos\left(\frac{4\pi}{7}\right)+\cos\left(\frac{6\pi}{7}\right) =-\frac{1}{2}$$

$$\cos\left(\frac{2\pi}{7}\right)+\cos\left(\frac{4\pi}{7}\right)+\cos\left(\frac{8\pi}{7}\right) =-\frac{1}{2}$$

$$8\sin\left(\frac{\pi}{8}\right)\sin\left(\frac{2\pi}{8}\right)=\frac{2\cdot{2}} {\sqrt{2+\sqrt{2}}}$$

$$8\sin\left(\frac{\pi}{10}\right)\sin\left(\frac{2\pi}{10}\right) \sin\left(\frac{3\pi}{10}\right)=\frac{\sqrt{2+\phi}}{\phi}$$

$$8\sin\left(\frac{\pi}{12}\right)\sin\left(\frac{2\pi}{12}\right) \sin\left(\frac{3\pi}{12}\right)\sin\left(\frac{4\pi}{12}\right) =\frac{\sqrt{3}}{1+\sqrt{3}}$$

$$8\cos\left(\frac{\pi}{8}\right)\cos\left(\frac{2\pi}{8}\right) \cos\left(\frac{3\pi}{8}\right)=2$$

$$8\cdot\cos\left(\frac{\pi}{5}\right)\cos\left(\frac{2\pi}{5}\right) \cos\left(\frac{3\pi}{5}\right)=-\frac{1}{\phi}$$

$$\left[8\cdot\sin\left(\frac{\pi}{5}\right)\sin\left(\frac{2\pi}{5}\right) \sin\left(\frac{3\pi}{5}\right)\right]^2-7=\phi^5$$

n ≥ 1

$$\tan\left(\frac{2\pi}{7}\right)\tan\left(\frac{3\pi}{7}\right)= -\tan\left(\frac{2^{2^n}\pi}{7}\right)\tan\left(\frac{3^{2^n}\pi}{7}\right)$$

Power Radical
$$\sqrt{\frac{p-q}{p}}=\sqrt{\frac{p(p-q)}{[p(p-q)]^{2^0}}-(p-q)\sqrt{\frac{p(p-q)}{[p(p-q)]^{2^1}}- (p-q)\sqrt{\frac{p(p-q)}{[p(p-q)]^{2^2}}-(p-q)\sqrt{\frac{p(p-q)}{[p(p-q)]^{2^3}}-\cdots}}}}$$

$$\sqrt{\frac{p}{p-q}}=\sqrt{\frac{p(p-q)}{[p(p-q)]^{2^0}}+(p-q)\sqrt{\frac{p(p-q)}{[p(p-q)]^{2^1}}+ (p-q)\sqrt{\frac{p(p-q)}{[p(p-q)]^{2^2}}+(p-q)\sqrt{\frac{p(p-q)}{[p(p-q)]^{2^3}}+\cdots}}}}$$

Erdos's equation

$$4!+1=5^2\cdots(1)$$

$$\sqrt{\frac{5^2}{4!}}=\sqrt{\frac{5^2(4!)}{[5^2(4!)]^{2^0}}+\sqrt{\frac{5^2(4!)}{[p(p-1)]^{2^1}}+ \sqrt{\frac{5^2(4!)}{[5^2(4!)]^{2^2}}+\sqrt{\frac{5^2(4!)}{[5^2(4!)]^{2^3}}+\cdots}}}}$$

phi

$$\phi-\frac{1}{2}=\sqrt{\frac{20}{20^{2^0}}+\sqrt{\frac{20}{20^{2^1}}+ \sqrt{\frac{20}{20^{2^2}}+\sqrt{\frac{20}{20^{2^3}}+\cdots}}}}$$

Radical
$$\frac{k+\sqrt{k^2+4k}}{2}=\sqrt{k+k\sqrt{k+k\sqrt{k+\cdots}}}$$

$$r\times\frac{k+\sqrt{k^2+4k}}{2}=\sqrt{r^2k+rk\sqrt{r^2k+rk\sqrt{r^2k+\cdots}}}$$

r = k = 1 : Yields phi

$$\frac{1+\sqrt{5}}{2}=\phi=\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}$$

Oleg
$$J_{4}^4(e^{-\pi}) \left[\frac{J_{4}^{3}(e^{-\pi})J_{4}^'(e^{-\pi})}{2\Psi^8(e^{-\pi})}+1\right]= \frac{1}{\pi}$$

Radical and phi
$$n^2(n+1)=\sqrt{\left[n(n+1)\right]^3- n(n+1)\sqrt{\left[n(n+1)\right]^3-n(n+1)\sqrt{\left[n(n+1)\right]^3-\cdots}}}$$

Special case of factorial

$$1!2!=\sqrt{2!^3-2!\sqrt{2!^3-2!\sqrt{2!^3-\cdots}}}$$

$$2!(3!)=\sqrt{3!^3-3!\sqrt{3!^3-3!\sqrt{3!^3-\cdots}}}$$

$$n(n+1)^2=\sqrt{\left[n(n+1)\right]^3+ n(n+1)\sqrt{\left[n(n+1)\right]^3+n(n+1)\sqrt{\left[n(n+1)\right]^3+\cdots}}}$$

Special case of factorial

$$2(2!)=\sqrt{2!^3+2!\sqrt{2!^3+2!\sqrt{2!^3+\cdots}}}$$

$$3(3!)=\sqrt{3!^3+3!\sqrt{3!^3+3!\sqrt{3!^3+\cdots}}}$$

$$\phi^{n+1}=\sqrt{\phi^{2n}+\phi^n\sqrt{\phi^{2n}+\phi^n\sqrt{\phi^{2n}+\cdots}}}$$

Square

$$\phi^{1}=\sqrt{\phi^{0^2}+\phi^0\sqrt{\phi^{0^2}+\phi^0\sqrt{\phi^{0^2}+\cdots}}}$$

$$\phi^{3}=\sqrt{\phi^{2^2}+\phi^2\sqrt{\phi^{2^2}+\phi^2\sqrt{\phi^{2^2}+\cdots}}}$$

$$\phi^{5}=\sqrt{\phi^{3^2}+\phi^3\sqrt{\phi^{3^2n}+\phi^2\sqrt{\phi^{3^2}+\cdots}}}$$

Continued fraction
$$\frac{1}{e^{\frac{1}{k}}-1}=1k-1+\frac{1k}{2k-1+\frac{2k}{3k-1+\frac{3k}{4k-1 +\frac{4k}{5k-1+\cdots}}}}$$

k = 1 : Yields Ramanujan's equation

$$\frac{1}{e^{\frac{1}{1}}-1}=0+\frac{1}{1+\frac{2}{2+\frac{3}{3 +\frac{4}{4+\cdots}}}}$$

Continued fraction factorial
$$3!=2!+\frac{4!}{2!+\frac{4!}{2!+\frac{4!}{2+\cdots}}}$$

$$(3!)^2-3!=3!+\frac{(3!)!}{3!+\frac{(3!)!}{3!+\frac{(3!)!}{3!+\cdots}}}$$

Phi
$$1\phi=\sqrt{1^1+1\sqrt{1^1+{1\sqrt{1^1+\cdots}}}}$$

$$2\phi=\sqrt{2^2+2\sqrt{2^2+{2\sqrt{2^2+\cdots}}}}$$

$$k\phi=\sqrt{k^2+k\sqrt{k^2+{k\sqrt{k^2+\cdots}}}}$$

Power and Factorial
$$P=\sqrt{k^k+k!\sqrt{k^k+{k!\sqrt{k^k+\cdots}}}}$$

$$1\phi=\sqrt{1^1+1!\sqrt{1^1+{1!\sqrt{1^1+\cdots}}}}$$

$$2\phi=\sqrt{2^2+2!\sqrt{2^2+{2!\sqrt{2^2+\cdots}}}}$$

$$3^2=\sqrt{3^3+3!\sqrt{3^3+{3!\sqrt{3^3+\cdots}}}}$$

$$2^{5}=\sqrt{4^4+4!\sqrt{4^4+{4!\sqrt{4^4+\cdots}}}}$$

Erdos's equation
$$4!+1=5^2\cdots(1)$$

$$5!+1=11^2\cdots(2)$$

$$7!+1=72^2\cdots(3)$$

$$5^2=\sqrt{5^2+4!\sqrt{5^2+{4!\sqrt{5^2+\cdots}}}}$$

$$11^2=\sqrt{11^2+5!\sqrt{11^2+{5!\sqrt{11^2+\cdots}}}}$$

$$72^2=\sqrt{72^2+7!\sqrt{72^2+{7!\sqrt{72^2+\cdots}}}}$$

Continued fraction
1 1 2 3 5 8 13 ... 

F1 = 1, F2 = 1

n > 1

$$\frac{\left(\phi^{2n}-1\right)\left(\phi^{2n-2}-1\right)} {\left(\phi^{2n}+1\right)\left(\phi^{2n-2}+1\right)}= \frac{\ln{F_{2n-1}}}{2+\frac{\left(\ln{F_{2n-1}}\right)^2} {6+\frac{\left(\ln{F_{2n-1}}\right)^2}{10+\frac{\left(\ln{F_{2n-1}}\right)^2}{16+\cdots}}}}$$

Radical
Phi

$$\phi^{\frac{1}{2^n}}=\sqrt{1+\frac{1}{\phi^{\frac{2^k+1}{2^k}}\prod_{k=1}^{n-1} \left(\phi^{\frac{1}{2^k}}+1\right)} \sqrt{1+\frac{1}{\phi^{\frac{2^k+1}{2^k}}\prod_{k=1}^{n-1}\left(\phi^{\frac{1}{2^k}}+1\right)} \sqrt{1+\frac{1}{\phi^{\frac{2^k+1}{2^k}}\prod_{k=1}^{n-1}\left(\phi^{\frac{1}{2^k}}+1\right)} \sqrt{1+\cdots}}}}$$

n = 1 : Yields

$$\sqrt{\phi}=\sqrt{1+\frac{1}{\phi\sqrt{\phi}}\sqrt{1+\frac{1}{\phi\sqrt{\phi}} \sqrt{1+\frac{1}{\phi\sqrt{\phi}}\sqrt{1\cdots}}}}$$

 2 

$$2^{\frac{1}{2^n}}=\sqrt{1+\frac{1}{2^{\frac{1}{2^k}}\prod_{k=1}^{n-1} \left(2^{\frac{1}{2^k}}+1\right)} \sqrt{1+\frac{1}{2^{\frac{2^k+1}{2^k}}\prod_{k=1}^{n-1}\left(2^{\frac{1}{2^k}}+1\right)} \sqrt{1+\frac{1}{2^{\frac{2^k+1}{2^k}}\prod_{k=1}^{n-1}\left(2^{\frac{1}{2^k}}+1\right)} \sqrt{1+\cdots}}}}$$

n = 1 : Yields

$$\sqrt{2}=\sqrt{1+\frac{1}{\sqrt{2}}\sqrt{1+\frac{1}{\sqrt{2}} \sqrt{1+\frac{1}{\sqrt{2}}\sqrt{1\cdots}}}}$$

Similiar to Gosper
$$\lim_{k\to\infty}\prod_{n=1}^{2^k+2^{k-3}-1} \frac{\pi}{2\sin^{-1}\left(\frac{10^k-n}{10^k}\right)}=4^{\frac{1}{\pi}}$$

$$\lim_{k\to\infty}\prod_{n=1}^{2^k+2^{k-3}-1} \frac{\pi}{2\cos^{-1}\left(\frac{n}{10^k}\right)}=4^{\frac{1}{\pi}}$$

$$\lim_{k\to\infty}\prod_{n=1}^{F(k)} \frac{\pi}{4\tan^{-1}\left(\frac{10^k-n}{10^k}\right)}=4^{\frac{1}{\pi}}$$

f(1) = 3, f(2) = 11 , f(3) = 37 , f(k) = 

power of 2 problem
$$4!+2^3=2^5\cdots(1)$$

$$5!+2^3=2^7\cdots(2)$$

$$n!+2^3=2^k\cdots(3)$$

Is there nay more of this type?

Factorial Radical
P is prime number and W is a whole number

$$n!+m=P\cdots(1)$$

$$W=\sqrt{n!+m\sqrt{n!+m\sqrt{n!+m\sqrt{n!+\cdots}}}}$$

$$2^1=\sqrt{2!+\sqrt{2!+\sqrt{2!+\sqrt{2!+\cdots}}}}$$

$$2^2-1=\sqrt{3!+\sqrt{3!+\sqrt{3!+\sqrt{3!+\cdots}}}}$$

$$2^3-2=\sqrt{3!+5\sqrt{3!+5\sqrt{3!+5\sqrt{3!+\cdots}}}}$$

$$2^3=\sqrt{4!+5\sqrt{4!+5\sqrt{4!+5\sqrt{4!+\cdots}}}}$$

$$2^4-1=\sqrt{5!+7\sqrt{5!+7\sqrt{5!+7\sqrt{5!+\cdots}}}}$$

$$2^5-2^3=\sqrt{6!+19\sqrt{6!+19\sqrt{6!+19\sqrt{6!+\cdots}}}}$$

$$2^7-2^4=\sqrt{7!+67\sqrt{7!+67\sqrt{7!+67\sqrt{7!+\cdots}}}}$$

Radical of phi
Equation:

$$x^{2n}-Ax^n-1=0\cdots(1)$$

$$x^n=\sqrt{1+A\sqrt{1+A\sqrt{1+A\sqrt{1\cdots}}}}$$

Radical:

$$\phi=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1\cdots}}}}$$

$$\phi=\sqrt{-1+\sqrt{5}\sqrt{-1+\sqrt{5}\sqrt{-1+\sqrt{5}\sqrt{-1\cdots}}}}$$

Equation:

$$\phi^2-\phi-1=0\cdots(2)$$

Radical:

$$\phi^3=\sqrt{1+4\sqrt{1+4\sqrt{1+4\sqrt{1\cdots}}}}$$

$$\phi^6-4\phi^3-1=0\cdots(4)$$

Radical:

$$\phi^5=\sqrt{1+11\sqrt{1+11\sqrt{1+11\sqrt{1\cdots}}}}$$

$$\phi^{10}-11\phi^5-1=0\cdots(6)$$

General Lucas Radical

'''Lucas's number: 2 1 3 4 7 11 18 ... Ln'''

'''L1 = 2, L2 = 1 , ... '''

$$\phi^{2n-1}=\sqrt{1+L_{2n}\sqrt{1+L_{2n}\sqrt{1+L_{2n}\sqrt{1\cdots}}}}$$

$$\phi^{4n-2}-L_{2n}\phi^{2n-1}-1=0\cdots(8)$$

$${\phi}=1+\frac{1}{1+\frac{1}{1+\frac{1}{1+...}}}$$

$${\phi^{2n-1}}=L_{2n}+\frac{1}{L_{2n}+ \frac{1}{L_{2n}+\frac{1}{L_{2n}+...}}}$$

General Fibonacci Radical

'''Fibonacci's number: 1 1 2 3 5 8 13 ... Fn'''

'''F1 = 1, F2 = 1 , ... '''

$$\phi^{2n}=\sqrt{1+F_{2n}\sqrt{5}\sqrt{1+F_{2n}\sqrt{5}\sqrt{1+F_{2n}\sqrt{5}\sqrt{1\cdots}}}}$$

$$\phi^{4n}-F_{2n}\sqrt{5}\phi^{2n}-1=0\cdots(10)$$

Example

$$\phi^{2}=\sqrt{1+\sqrt{5}\sqrt{1+\sqrt{5}\sqrt{1+\sqrt{5}\sqrt{1\cdots}}}}$$

Continued fraction

$${\phi^2}=\sqrt{5}+\frac{1}{\sqrt{5}+\frac{1}{\sqrt{5}+\frac{1}{\sqrt{5}+...}}}$$

$${\phi^{2n}}=F_{2n}\sqrt{5}+\frac{1}{F_{2n}\sqrt{5}+ \frac{1}{F_{2n}\sqrt{5}+\frac{1}{F_{2n}\sqrt{5}+...}}}$$

Almost An Integer
Let C and D be any inetgers 

$$\lim_{n\to\infty}\sqrt{2^{2n}\pm{2}}\times\left(2^{2n+1}\pm{1}\right)=C\cdots(1)$$

C is always divisible by

$$2^n\cdots(2)$$

$$\sqrt{2^{8}-2}\times\left(2^{9}-1\right)=8143.999877...\cdots(1)$$

$$\frac{8144}{2^4}=509\cdots(2)$$

Odd number

$$\lim_{n\to\infty}\sqrt{2^{2n}\pm{2}}\times\left(2^{n+1}\pm{2^{-n}}\right)=D$$

D is always an odd number

Sum and Difference

$$\lim_{n\to\infty}\sqrt{2^{2n}+{2}}\times\left(2^{n+1}+2^{-n}\right)- \lim_{n\to\infty}\sqrt{2^{2n}-{2}}\times\left(2^{n+1}-2^{-n}\right)=6$$

$$\lim_{n\to\infty}\sqrt{2^{2n}+{2}}\times\left(2^{n+1}+2^{-n}\right)+ \lim_{n\to\infty}\sqrt{2^{2n}-{2}}\times\left(2^{n+1}-2^{-n}\right)=2^{2n+2}$$

General

$$\lim_{n\to\infty}\sqrt{k^{2n}+{k}}\times\left(k^{n+1}+k^{-n}\right)- \lim_{n\to\infty}\sqrt{k^{2n}-{k}}\times\left(k^{n+1}-2^{-k}\right)=k^2+2$$

$$\lim_{n\to\infty}\sqrt{k^{2n}+{k}}\times\left(k^{n+1}+k^{-n}\right)+ \lim_{n\to\infty}\sqrt{k^{2n}-{k}}\times\left(k^{n+1}-k^{-n}\right)=2k^{2n+1}$$

General

$$\lim_{n\to\infty}\sqrt{k^{2n}\pm{k}}\times\left(k^{n+1}\pm{k^{-n}}\right)- k^{2n+1}=\pm{\frac{k^2+2}{2}}$$

Mixed
$$\sum_{n=1}^{\infty}\frac{(-1)^{n+1}2^{2n}B_{2n}}{2(2n+1)(2n)!} =\ln\prod_{n=1}^{\infty}\left(\frac{n\pi+1}{n\pi-1}\right)^{\frac{n\pi}{2}} \frac{1}{e}$$

$$\sum_{n=1}^{\infty}\frac{(-1)^{n+1}2^{2n}B_{2n}}{n(2n+1)(2n)!} =\ln\frac{1}{\sin^2\left(\frac{180}{\pi}\right)}\prod_{n=1}^{\infty}\left(\frac{n\pi-1}{n\pi+1}\right)^{2n\pi} {e^4}$$

Strange square number
Ignore the number inside the [ ]

$$36^2=1[2]96=14^2\cdots(1)$$

$$37^2=1[3]69=13^2\cdots(2)$$

$$38^2=1[4]44=12^2\cdots(3)$$

$$39^2=1[5]21=11^2\cdots(4)$$

$$40^2=1[6]00=10^2\cdots(5)$$

Other example

$$14^2=1[9]6=4^2\cdots(1)$$

$$15^2=2[2]5=5^2\cdots(2)$$

Limit
$$\lim_{n\to\infty}\left(\frac{kn+a}{kn-b}\right)^{xn}=e^{\frac{(a+b)x}{k}}$$

Infinite product
$$\frac{e^{1-\frac{9\sqrt{3}}{4\pi}L_{-3}\left(2\right)}}{\sqrt{1}}= \prod_{n=1}^{\infty}\frac{1}{e^2}\left(\frac{6n+1}{6n-1}\right)^{6n}$$

$$\frac{e^{1-\frac{2K}{\pi}}}{\sqrt{2}}= \prod_{n=1}^{\infty}\frac{1}{e^2}\left(\frac{4n+1}{4n-1}\right)^{4n}$$

$$\frac{e^\left(1+{\frac{2\pi^2-3\psi_1\left(\frac{1}{3}\right)}{6\pi\sqrt{3}}}\right)} {\sqrt{3}}= \prod_{n=1}^{\infty}\frac{1}{e^2}\left(\frac{3n+1}{3n-1}\right)^{3n}$$

$$\frac{e}{\sqrt{4}}=\prod_{n=1}^{\infty}\frac{1}{e^2}\left(\frac{2n+1}{2n-1}\right)^{2n}$$

$$\frac{3^{\frac{7}{24}}\Gamma^{\frac{1}{2}}\left(\frac{1}{3}\right)} {A^4}e^{-\left({\frac{2\pi^2-3\psi_1\left(\frac{1}{3}\right)}{12\pi\sqrt{3}}}\right)}= \prod_{n=1}^{\infty}e\left(1-\frac{1}{3n}\right)^{3n-\frac{1}{2}}$$

$$\frac{2^{\frac{1}{12}}e}{A^3\pi^{\frac{1}{4}}}=\prod_{n=1}^{\infty}\frac{1}{e} \left(1+\frac{1}{2n}\right)^{2n+\frac{1}{2}}$$

$$\frac{2^{\frac{7}{12}}\pi^{\frac{1}{4}}} {A^3} =\prod_{n=1}^{\infty}{e} \left(1-\frac{1}{2n}\right)^{2n-\frac{1}{2}}$$

Barnes G function
$$G\left(\frac{1}{3}\right)G\left(\frac{2}{3}\right) G\left(\frac{3}{3}\right)G\left(\frac{4}{3}\right)= \frac{3^{\frac{5}{24}}}{\left(2\pi\right)^{\frac{1}{3}}A^{4}} e^{\frac{1}{9}\left(3+\frac{2\pi^2-3\psi_{1}\left(\frac{1}{3}\right)}{4\pi\sqrt{3}}\right)}$$

Fibonacci and phi series
1     1      2     3      5     8     13     21 ...   

''F1    F2     F3                              ... Fn''

$$\sum_{n=1}^{\infty}\frac{(-1)^{n+1}F_n}{F_{n+1}F_{n+2}}= \sum_{n=1}^{\infty}\frac{(-1)^{n+1}F_{n+1}}{F_n}$$

$$\frac{1}{F_{k+1}F_{k+2}}=\sum_{n=1}^{\infty}\frac{1}{F_{n+k}F_{n+k+2}}$$

$$(-1)^{k+1}\frac{2F_{2k+3}}{F_{2k+3}-F^2_{k}}+(-1)^k\sqrt{5}= \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{F_{n+k}F_{n+k+2}}$$

$$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{2F_{2n+3}}{F_{2n+3}-F^2_{n}}=\frac{11}{5}-\phi$$

Curiosity
$$\sqrt{3^3+3^2\sqrt{3^3+3^2}}=3^2$$

$$\sqrt{8^3+8^2}=24^2$$

$$(3!)!+4!=744$$

Fibonacci inside the [ ] 

$$\sqrt{1^2+12^2+123^2}=[123.58]8025...$$

$$4!+4=5^2-3\cdots(1)$$

$$5!+5=27^2-3\cdots(2)$$

$$6^4=6^2+(666)^1+(666)^1\cdots(1)$$

$$\frac{1188}{3}=396$$

$$\left(6\times6+6\times6-6\right)\times6=396$$

$$\sqrt{1^2+396\sqrt{2^2+396}}=89$$

$$1103+26390+396=167^2\cdots(1)$$

Almost Integers And Approximation
$$10e-\left(1+\frac{e}{\phi}\right)^2\approx{20}$$

$$\frac{36}{\sqrt{6}}+\frac{7}{\sqrt[4]7}\approx{19}$$

$$8\left(\frac{1}{4^3}+\frac{1}{10^3}+\frac{1}{18^3}\right)+\pi^2-10 \approx{\frac{1}{251.499996}}$$

$$\phi^{\frac{2}{3}\phi^{\sqrt{7}}}-\frac{1}{2^8}\approx{\pi}$$

$$\left(\frac{10\phi^3\sqrt{3-\sqrt{6}}-10\pi}{\pi-3}+\frac{\pi}{10}+1\right)^2\approx{1.9999}$$

$$\pi\approx{\sqrt{3^2+\left(\frac{2}{3}\right)^3\sqrt{3^2+\left(\frac{2}{3}\right)^4}}} -\frac{1}{(2\times3)^3}+\frac{1}{(3\times5)^4}$$

$$\left(\frac{e^{\pi}+2}{e^{\pi}-2}\right)^{2^2}\approx{2}$$

$$\left(\frac{1}{3-e}\right)^2\approx{\frac{5^3+1}{3^2+1}}$$

$$\frac{\phi^{2\sqrt{3}}-1}{\phi^\sqrt{3}}\approx{1+\frac{\sqrt{3}}{2}}$$

$$\frac{7!}{2\times\left(\ln2\right)^7\times{7^2}}\approx{669}$$

$$\frac{3}{1\times\left(\ln2\right)^{4e}}\approx{\left(\frac{7}{8-5}\right)^6}$$

$$\frac{7!}{\left[\ln\left(8\right)\right]^9\times10}\approx{\ln2}$$

$$\frac{6!}{\left[\ln\left(7\right)\right]^8\times9}\approx{\frac{3+4}{1+2+5+10}}$$

$$4\left(\ln4\right)^{4^2}-\frac{4+1}{4^2+1}\approx{744}$$

$$3\left(\ln3\right)^{3^2}\approx{7}$$

$$10^8\left(\frac{\sqrt{89}}{\pi}-\left(\frac{5}{6}\right)^{32}-3\right)- \frac{8+2}{\pi+e^{\frac{396}{8^2+2}+\frac{8}{396+\sqrt{9801}}}}\approx{396}$$

$$10^8\left(\frac{\sqrt{\sqrt{9801}-(8+2)}}{\pi}- \left[\left[2\left(\sqrt{2}\right)^2+1\right]\times\frac{8^2+2}{396}\right]^{32}- 2\left(\sqrt{2}\right)^2+1\right)- \frac{8+2}{\pi+e^{\frac{396}{8^2+2}+\frac{8}{396+\sqrt{9801}}}}\approx{396}$$

$$\frac{10}{A^4}\sqrt{\frac{\Gamma\left(\frac{1}{3}\right)}{2\pi}}-1\approx{\sqrt{2}}$$

$$\sqrt{8}\approx{\ln\frac{167}{\pi^2}}$$

$$\ln\left(1103\times26390\times396\right)\approx{e^{\pi\left(1+\frac{\pi^2}{26390}\right)}+ e^{-\pi^2}}$$

$$396\approx\frac{1}{5+\phi}+e^{\left({\pi}\sqrt{\frac{2\pi}{\sqrt{3}}}-\frac{1}{396}\right)}$$

$$\sqrt{2}\approx\pi^2\left(9801+4^{\frac{8}{\pi}}\right)e^{-2\pi^{1.5}}+e^{-e^{1.5^2}}$$

Other unit fractions
$$\frac{\pi}{2}=2\tan^{-1}\left(\frac{1}{2^{\frac{1}{2^{r+1}}}}\right)+ \tan^{-1}\left[\frac{1}{\left(\sqrt{2}+1\right)2^{\frac{2^{r+1}+1}{2^{r+1}}}\prod_{m=1}^{r-1} \left(2^{\frac{1}{2^{m+1}}}+1\right)}\right]$$

$$\frac{\pi}{2}=2\tan^{-1}\left(\frac{1}{3^{\frac{1}{2^{r+1}}}}\right)+ \tan^{-1}\left[\frac{1}{\left(\sqrt{3}+1\right)3^{\frac{1}{2^{r+1}}}\prod_{m=1}^{r-1} \left(3^{\frac{1}{2^{m+1}}}+1\right)}\right]$$

$$\frac{\pi}{2}=2\tan^{-1}\left(\frac{1}{5^{\frac{1}{2^{r+1}}}}\right)+ \tan^{-1}\left[\frac{1}{5^{\frac{1}{2^{r+1}}}{\phi}\prod_{m=1}^{r-1} \left(5^{\frac{1}{2^{m+1}}}+1\right)}\right]$$

$$\frac{\pi}{2}=2\tan^{-1}\left(\frac{1}{\phi^{\frac{1}{2^r}}}\right)+ \tan^{-1}\left[\frac{1}{\phi^{\frac{2^r+1}{2^r}}\prod_{m=1}^{r-1} \left(\phi^{\frac{1}{2^{m}}}+1\right)}\right]$$