User:Mathsformulae

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Dr. Ron Knott

Arctan relating to Pi, Phi and Fibonacci

$$\frac{\pi}{4}=arctan\left(\frac{1}{\sqrt{3}+1}\right)+arctan\left(\frac{1}{3\sqrt{3}+6}\right) +2arctan\left(\frac{1}{\sqrt{3}+4}\right)\cdots(1)$$

$$\frac{\pi}{4}=arctan\left(\frac{1}{nF_{2p-1}+1}\right) +\sum_{r=1}^narctan\left[{\frac{1}{r^2F_{2p-1}+r(2-F_{2p-1}) +\frac{2-F_{2p-1}}{F_{2p-1}}}}\right]\cdots(2)$$

$$\frac{\pi}{4}=arctan\left(\frac{1}{2\cdot1^2}\right)+arctan\left(\frac{1}{2\cdot2^2}\right)+arctan\left(\frac{1}{2\cdot3^2}\right)+\cdots+\arctan\left(\frac{1}{2r^2}\right)+ arctan\left(\frac{1}{2r+1}\right)\cdots(3)$$

$$\frac{\pi}{4}=\sum_{r=1}^{\infty}arctan\left({\frac{1}{2r^2}}\right)\cdots(4)$$

$$\frac{\pi}{4}=\sum_{r=1}^{\infty}arctan\left({\frac{1}{r^2+r+1}}\right)\cdots(5)$$

Special conditions r = 1  yield,

$$\frac{\pi}{4}=arctan\left(\frac{1}{2}\right)+arctan\left(\frac{1}{3}\right)$$which is Euler formula r = 2 gives,

$$\frac{\pi}{4}=arctan\left(\frac{1}{2}\right)+arctan\left(\frac{1}{8}\right)+arctan\left(\frac{1}{5}\right)$$which is Daze formula

$$\frac{\pi}{2}= arctan\left(\frac{2}{1^2}\right)+arctan\left(\frac{2}{3^2}\right)+arctan\left(\frac{2}{5^2}\right) +arctan\left(\frac{2}{7^2}\right)+\cdots(6)$$

$$arctan\left(\frac{1}{xF_{2n-1}+F_{2n}}\right)= arctan\left(\frac{1}{{x}F_{2n-1}+F_{2n+1}}\right)+arctan\left(\frac{1}{{x^2}F_{2n-1}+{x}F_{2n+2}+F_{2n+2}}\right)\cdots(7) $$

General

$$arctan\left[\frac{1}{kF_{2p-1}+zF_{2p-1}+F_{2p}}\right]=arctan\left[\frac{1}{(k+n)F_{2p-1}+zF_{2p-1}+F_{2p}}\right]+\sum_{r=1}^n arctan\left[{\frac{1}{(r+k+z)^2F_{2p-1}+(k+r+z)L_{2p}+ \frac{F_{2p-2}F_{2p}+1}{F_{2p-1}}}}\right]\cdots(8)$$

$$arctan\left(\frac{1}{kF_{2p-1}+L_{2p-1}}\right)=arctan\left(\frac{1}{(k+n)F_{2p-1}+L_{2p-1}}\right)+ \sum_{r=1}^narctan\left({\frac{1}{(r+k)^2F_{2p-1}+ \frac{1}{2}(r+k)(L_{2p+1}-F_{2p-6})+L_{2p-3}}}\right)\cdots(9)$$

$$arctan\left(\frac{1}{kF_{2p-1}-F_{2p-3}}\right)=arctan\left(\frac{1}{(k+n)F_{2p-1}-F_{2p-3}}\right)+ \sum_{r=1}^narctan\left({\frac{1}{(r+k)^2F_{2p-1}- \frac{1}{2}(r+k)(L_{2p+1}-F_{2p-6})+L_{2p-3}}}\right)\cdots(10)$$

Fibonacci Identities

N_ 1 2 3 4 5 ...

Fn: 1 1 2 3 5 ...

Ln: 2 1 3 4 7 ...

$$L_{n+2}F_{n+2}^2+L_{n+3}F_{n+3}^2=L_{3n+6}+L_{n+1}F_{n+1}^2\cdots(11)$$(F1=1,L1=1)

$$F_{n+3}^2+F_{n+4}^2=L_{2n+5}+F_{n}F_{n+1}+F_{n+3}F_{n+4}\cdots(12)$$

$$F_{n}F_{n+1}+F_{n+2}^2=F_{2n+2}+F_{n+1}^2+(-1)^{n+1}\cdots(13)$$

$$F_{n}F_{n+1}+2F_{n+2}^2=F_{2n+4}+(-1)^{n+1}\cdots(14)$$

$$F_{n}F_{n+1}+F_{n+2}^2=F_{2n+3}+F_{n}^2-1\cdots(15)$$

$$4L_{n}L_{n+1}+L_{n-1}^2=L_{n+2}^2\cdots(16)$$

$$4F_{n}F_{n+1}+F_{n-1}^2=F_{n+2}^2\cdots(17)$$

$$F_{2n+3}+1=F_{n}^2+F_{n+2}^2+F_{n}F_{n+1}\cdots(18)$$--Where n is odd

$$F_{2n+3}+1=F_{n}^2+F_{n+2}^2+F_{n}F_{n+1}+2\cdots(19)$$--Where n is even

$$F_{n+2}F_{n+6}-F_{n}F_{n+4}=F_{2n+6}\cdots(20)$$

$$F_{n+2}F_{n+3}-F_{n}F_{n+1}=F_{2n+3}\cdots(21)$$

$$F_{2n+1}^2+F_{2n+3}^2=F_{2n+2}(F_{2n}+F_{2n+4})+2\cdots(22)$$

$$F_{2n}^2+F_{2n+2}^2+2=F_{2n+1}(F_{2n-1}+F_{2n+3})\cdots(23)$$

$$F_{n}^3+F_{n+k}^3=(F_{n}+F_{n+k})^2+(F_{n}+F_{n+k})(F_{n+k}-F_{n})^2\cdots(24)$$

$$F_{n+3}F_{n+4}+2F_{n}F_{n+2}=L_{n+3}L_{n+4}+L_{2n+7}\cdots(25)$$

$$5F_{2n+6}=F_{n+5}F_{n+6}+F_{n}F_{n+1}\cdots(26)$$

$$F_{n+5}F_{n+6}+F_{n+8}^2=L_{2n+14}-5F_{2n+6}+F_{n}F_{n+1}\cdots(27)$$

$${F_{n+1}+4F_{n+2}+9F_{n+3}+16F_{n+4}+25F_{n+5}+36F_{n+6}}= F_{n+13}+F_{n+10}+F_{n+8}+F_{n+5}+F_{n+3}+F_{n}\cdots(28)$$

$$F_{n+1}L_{n+2}+F_{n+2}^2+F_{n+3}^2=2F_{2n+4}\cdots(29)$$

$$F_{2n}^2=F_{2n-1}F_{2n}+F_{2n-2}^2-1\cdots(30)$$

$$F_{2n+1}^2=F_{2n}F_{2n+1}+F_{2n}^2+1\cdots(31)$$

$$F_{n+4}^2F_{n+3}+F_{n+1}F_{n}^2=L_{3n+7}\cdots(32)$$

$$F_{n}^5+F_{n+1}^5=F_{n+2}[(F_{n}F_{n+1}+F_{n-1}^2)^2+F_{n-1}^2F_{n}F_{n+1}]\cdots(33)$$

$$F_{n}^5+L_{n}^5=(F_{n}+L_{n})[[F_{n}L_{n}+(L_{n}-F_{n})^2]^2+(L_{n}-F_{n})^2F_{n}L_{n}]\cdots(34)$$

$$\frac{F_{n-1}^3+L_{n+1}^3}{F_{n+1}+L_{n+1}}=4F_{2n-1}-F_{n}L_{n}-F_{n-2}^2\cdots(35)$$

$$F_{n}^3+L_{n}^3=(F_{n}+L_{n})(F_{n-2}^2+F_{n}L_{n})\cdots(36)$$

$$F_{n+2}^4-F_{n+1}^4=F_{n}F_{n+3}F_{2n+3}\cdots(37)$$

$$F_{n+3}^3-F_{n+2}^3=F_{n+1}(F_{n}F_{n+1}+L_{3n})\cdots(38)$$

$$F_{n+4}^2-F_{n}^2=3F_{n+2}L_{n+3}\cdots(39)$$

$$F_{n+3}^2-F_{n}^2=4F_{n+1}F_{n+2}\cdots(40)$$

$$F_{n+2}^2-F_{n}^2=F_{n+1}L_{n+2}\cdots(41)$$

$$F_{n}^2+F_{n+1}^2+L_{n}^2=4L_{2n-1}+2(-1)^{n+1}\cdots(42)$$

$$F_{n}^2+F_{n+1}^2=2L_{n-1}F_{n}+L_{n}^2\cdots(43)$$

$$2(F_{n}^2+F_{n+1}^2)-(L_{n}^2+L_{n+1}^2)=F_{2n-4}\cdots(44)$$

$$\sqrt{F_{2n+1}F_{2n+2}}=\sqrt{F_{2n}F_{2n+3}}+\frac{1}{2\sqrt{F_{2n}F_{2n+3}}}\cdots(45)$$

$$F_{n+1}(L_{n+3})+L_{n+1}(F_{n+3})=F_{2n-2}+F_{2n+4}\cdots(46)$$

$$L_{n}^3+L_{n+1}^3=F_{n+2}(L_{n+1}^2+L_{2n-5})+(-1)^{n+1}(L_{n+1}+F_{n-7})\cdots(47)$$

$$L_{n}^4+L_{n+1}^4+L_{n+2}^4=2(L_{n}L_{n+2}+L_{n}^2)^2\cdots(48)$$

$$L_{n-2k}L_{n+2k+1}-L_{n-2k-1}L_{n+2k+2}=5(-1)^{k+1}F_{2k+2}\cdots(49)$$

$$L_{n}L_{n+1}^2=5L_{n}+L_{n-1}L_{n}^2+2L_{n}^3\cdots(50)$$

$$L_{2n+2}^2-L_{2n}^2=5F_{4n}^2\cdots(51)$$

$$L_{2n+3}^2-L_{2n+1}^2=5F_{4n+2}^2\cdots(52)$$

$$L_{n}^2+L_{n+1}^2=F_{n+2}^2+F_{n-3}^2\cdots(53)$$

$$\frac{F_{n+1}^3+F_{n+2}^3}{F_{n+3}}+\frac{F_{n+2}^3+F_{n+3}^3}{F_{n+4}}=\frac{F_{n}^3+F_{n+3}^3}{2F_{n+2}}+F_{n}F_{n+3}\cdots(54)$$

$$\frac{F_{2n+1}^2+F_{2n+3}^2}{F_{2n+2}}+\frac{F_{2n+2}^2+F_{2n+4}^2}{F_{2n+3}}-\frac{F_{2n+3}^2+F_{2n+5}^2}{F_{2n+4}}+1 =\frac{2}{F_{2n+2}}+\frac{{F_{2n+3}}-2}{F_{2n+3}}-\frac{2}{F_{2n+4}}\cdots(55)$$

$$\frac{F_{2n}^2+F_{2n+2}^2}{F_{2n+1}}+\frac{F_{2n+1}^2+F_{2n+3}^2}{F_{2n+2}}-\frac{F_{2n+2}^2+F_{2n+4}^2}{F_{2n+3}} =\frac{F_{2n+1}-2}{F_{2n+1}}+\frac{2}{F_{2n+2}}-\frac{F_{2n+3}-2}{F_{2n+3}}\cdots(56)$$

$$F_{n+2}(F_{n+1}+F_{n+5})=F_{n+3}(F_{n}+F_{n+4})\cdots(57)$$

$$F_{2n-1}F_{2n+2m-1}=F_{m}^2+F_{2n+m-1}^2\cdots(58)$$

$$ F_{n}F_{n+r}-F_{n+r-2}F_{n-2}=(-1)^nF_{r}\cdots(59)$$

$$ 3(-1)^n[F_{3n}-L_{n+1}^2F_{n}] =F_{3n-1}-L_{n}^2F_{n+1}\cdots(60)\,$$

Cube sum Of Fibonacci and Lucas series

Fibonacci series

$$S_{F}=1^3+1^3+2^3+3^3+5^3+8^3+...+F_{n+k-1}^3$$

Fibonacci cube sum formula

$$S_{F}=\frac{F_{n+k-1}F_{n+k}^2-F_{n-1}F_{n}^2+(-1)^{n+k}[F_{n+k-2}+(-1)^{k+1}F_{n-2}]}{2}\cdots(62)$$

Lucas series

$$S_{L}=2^3+1^3+3^3+4^3+7^3+11^3+...+L_{n+k-1}^3$$

Lucas cube sum formula

$$S_{L}=\frac{L_{n+k-1}L_{n+k}^2-L_{n-1}L_{n}^2+5(-1)^{n+k}[L_{n+k-2}+(-1)^{k+1}L_{n-2}]}{2}\cdots(63)$$

$$S_{L}=2^3+1^3+3^3+4^3+7^3\cdots+L_{n}$$

$$S_{L}=L_{n}^2L_{n+1}-L_2L_{3}L_{4}-L_3L_{4}L_{5}-L_4L_{5}L_{6}-\cdots-L_{n-2}L_{n-1}L_n\cdots(64)$$

$$S_{F_{m,n}}={1+1+2+3+5+8+...+F_{m}+F_{n}}$$

$$S_{F_{m,n}}=\left(\frac{\sqrt{5}+5}{10}\right)\left[\left(\frac{1+\sqrt{5}}{2}\right)^{m+1}-\left(\frac{1+\sqrt{5}}{2}\right)^{n}\right] -\left(\frac{\sqrt{5}-5}{10}\right)\left[\left(\frac{1-\sqrt{5}}{2}\right)^{m+1}-\left(\frac{1-\sqrt{5}}{2}\right)^{n}\right]\cdots(65)$$

Pythagoras Fibonacci

$${(F_{2n}F_{2n+3}+F_{2n+1}F_{2n+2}-1)}^2+[F_{2n+2}(F_{2n}F_{2n+1}-2)]^2= [F_{2n+1}(F_{2n}F_{2n+2}+2)]^2\cdots(66)$$

$${(F_{n}F_{n+k}-1)}^2+(F_{n}+F_{n+k})^2=(F_{n}F_{n+k}+1)^2\cdots(67)$$

$${(F_{2n-1}F_{2n}F_{2n+1}-2F_{2n+1})}^2+(F_{2n-1}F_{2n}+F_{2n+1}^2-1)^2=2^2+ (F_{2n-1}F_{2n}F_{2n+1}+2F_{2n})^2\cdots(68)$$

Fibonacci and Phi

$$\frac{F_{n-1}+1}{F_{n-1}-1}=\left(\frac{\phi^n+1}{\phi^n-1}\right)\left(\frac{{\phi}^{n-2}+1}{{\phi}^{n-2}-1}\right)\cdots(69)$$

Phi relating to Fibonacci Lucas numbers

$$\frac=\frac{L_{n+2}}{L_{n+1}}\cdots(70)$$

Where n is odd

$$\frac=\frac{F_{n+1}}{F_{n}}\cdots(71)$$

Where is even

$$\frac{1+{\phi^{8n}}}{{\phi}^{2n}+{\phi}^{6n}}=\frac{L_{2n}^2-2}{L_{2n}}\cdots(72)$$

$$\frac{1+{\phi^{10n}}}{{\phi}^{2n}+{\phi}^{8n}}=\frac{L_{2n}^2-L_{2n}-1}{L_{2n}-1}\cdots(73)$$

$$\frac{1+{\phi^{12n}}}{{\phi}^{2n}+{\phi}^{10n}}=\frac{L_{2n}(L_{4n}-1)}{L_{4n}}\cdots(74)$$

$$\frac{1+{\phi^{8n}}}{{\phi}^{2n}+{\phi}^{4n}+{\phi^{6n}}}=\frac{L_{2n}^2-2}{L_{2n}+1}\cdots(75)$$

$$\frac{1+{\phi^{10n}}}{{\phi}^{2n}+{\phi}^{4n}+{\phi^{6n}}+{\phi^{8n}}}=\frac{L_{2n+1}^2-L_{2n+1}-1}{L_{2n+1}}\cdots(76)$$

Pi and Arctan series involve Fibonacci and Lucas numbers

$$\frac{\pi}{4}=arctan\left(\frac{1}{2}\right)+arctan\left(\frac{1}{18}\right)+ arctan\left(\frac{1}{34}\right)+arctan\left(\frac{1}{322}\right) +arctan\left(\frac{1}{610}\right)+\cdots$$

$$+arctan\left(\frac{4}{18}\right)+arctan\left(\frac{4}{322}\right) +arctan\left(\frac{4}{5778}\right)+arctan\left(\frac{4}{103682}\right)+\cdots(78)$$

$$\frac{\pi}{6}=arctan\left(\frac{1}{5}\right)+arctan\left(\frac{1}{8}\right)+ arctan\left(\frac{1}{13}\right)+arctan\left(\frac{1}{21}\right)+\cdots(79)$$

$$\frac{\pi}{4}=arctan\left(\frac{15L_{6}}{L_{6}^2-44}\right)+arctan\left(\frac{5L_{12}}{L_{12}^2-4}\right)+ arctan\left(\frac{15L_{18}}{L_{18}^2-44}\right) +arctan\left(\frac{5L_{24}}{L_{24}^2-4}\right)+\cdots(80)$$

$$\frac{\pi}{4}=arctan\left(\frac{3\sqrt{5}}{7}\right)+arctan\left(\frac{3\sqrt{5}}{322}\right)+ arctan\left(\frac{3\sqrt{5}}{15177}\right) +\cdots+arctan\left(\frac{3\sqrt{5}}{L_{n+8}}\right)+\cdots(81)$$

$$\frac{\pi}{4}=arctan\left(\frac{3}{\sqrt{5}}\right)+arctan\left(\frac{3}{8\sqrt{5}}\right) arctan\left(\frac{3}{55\sqrt{5}}\right)+arctan\left(\frac{3}{377\sqrt{5}}\right)+\cdots(82)$$

$$\frac{\pi}{4}=arctan\left(\frac{2\sqrt{5}}{4}\right)-arctan\left(\frac{2\sqrt{5}}{76}\right)+arctan\left(\frac{2\sqrt{5}}{1364}\right)-\cdots- arctan\left(\frac{2\sqrt{5}}{L_{6n-3}}\right)+\cdots(83)$$

$$\frac{\pi}{4}=arctan\left(\frac{1}{2}\right)+arctan\left(\frac{1}{4}\right)+arctan\left(\frac{1}{24}\right)+arctan\left(\frac{1}{29}\right) arctan\left(\frac{1}{1515}\right)+arctan\left(\frac{1}{6884163}\right)\cdots(84)$$

$$\frac{\pi}{4}=arctan\left(\frac{1}{2}\right)+2arctan\left(\frac{1}{6}\right)-arctan\left(\frac{1}{117}\right)\cdots(85)$$

$$\frac{\pi}{4}=arctan\left(\frac{1}{2}\right)+arctan\left(\frac{1}{4}\right)+arctan\left(\frac{1}{16}\right) +arctan\left(\frac{1}{70}\right)+arctan\left(\frac{1}{14633}\right)\cdots(86)$$

$$\frac{\pi}{4}=arctan\left(\frac{1}{2}\right)+arctan\left(\frac{1}{5}\right)+arctan\left(\frac{1}{10}\right) arctan\left(\frac{1}{41}\right)+arctan\left(\frac{1}{3323}\right)\cdots(87)$$

$$\frac{\pi}{4}=arctan\left(\frac{1}{2}\right)+arctan\left(\frac{1}{5}\right)+arctan\left(\frac{1}{12}\right) +arctan\left(\frac{1}{25}\right)+arctan\left(\frac{1}{810}\right)+arctan\left(\frac{1}{17493}\right)\cdots(88)$$

$$\frac{\pi}{4}=arctan\left(\frac{1}{2}\right)+arctan\left(\frac{1}{4}\right)+arctan\left(\frac{1}{14}\right) +arctan\left(\frac{1}{25}\right)+arctan\left(\frac{1}{41}\right) +arctan\left(\frac{1}{735}\right)+arctan\left(\frac{1}{1079717}\right)\cdots(89)$$

$$\frac{\pi}{4}=arctan\left(\frac{1}{2}\right)+arctan\left(\frac{1}{4}\right)+arctan\left(\frac{1}{22}\right) +arctan\left(\frac{1}{32}\right) +arctan\left(\frac{1}{9193}\right)\cdots(90)$$

Not relating to Fibonacci number

Definition of:

$$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}=1+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\cdots$$

$$\eta(s)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^s}=1-\frac{1}{2^s}+\frac{1}{3^s}-\frac{1}{4^s}+\cdots$$

$$\lambda(s)=\sum_{n=0}^{\infty}\frac{1}{(2n+1)^s}=1+\frac{1}{3^s}+\frac{1}{5^s}+\frac{1}{7^s}+\cdots$$

$$\beta(s)=\sum_{n=0}^{\infty}\frac{(-1)^{k}}{(2n+1)^s}=1-\frac{1}{3^s}+\frac{1}{5^s}-\frac{1}{7^s}+\cdots$$

Where p is prime numbers

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

Jacobi theta function

$$J_4(0,e^{-1})=\left(\frac{\pi}{2}\right)^{\frac{1}{4}}\cdot \frac{1}{\Gamma\left(\frac{3}{4}\right)}\cdots(1)$$

$$\frac{J_{2}(0,e^{-1})}{J_{1}^'(0,e^{-1})}=\left(\frac{\pi^2}{2} \right)^{\frac{1}{4}}\cdot\frac{1}{\Gamma^2\left(\frac{3}{4}\right)}\cdots(2)$$

Infinite product

$$\pi=\frac{\prod_{n=1}^{\infty}\left[1+\frac{1}{4(3n)^2-1}\right]}{\sum_{n=1}^{\infty}\frac{1}{4^n} }\cdots(1)$$

$$\frac{\Gamma^2(\frac{1}{4})}{\pi^{\frac{3}{2}}}=\frac{\prod_{n=1}^{\infty}\left[1+\frac{1}{(4n-1)^2-1}\right]}{\sum_{n=1}^{\infty}\frac{1}{4n^2-1}}\cdots(2)$$

$$\frac{\Gamma^2(\frac{1}{4})}{\pi^{\frac{5}{2}}}={\sum_{n=1}^{\infty}\frac{1}{4n^2-1}}\cdot \cdots(3)$$

$$\left[\frac{\Gamma^(\frac{1}{4})}{2}\right]^2\cdot{\frac{1}{\sqrt{2\pi}}}=\frac{3}{2}\cdot\frac{4}{5}\cdot\frac{7}{6}\cdot\frac{8}{9} \cdot\frac{11}{10}\cdot\frac{12}{13}=\prod_{k=2,4,6,...}^{\infty}\left(\frac{k+1}{k}\right)^{(-1)^{\frac{k}{2}+1}} \cdots(4)$$

$$\frac{4}{\pi}=\lim_{n\to\infty}(4n+1)\cdot\left(\frac{1}{2}\cdot \frac{3}{4}\cdot\frac{5}{6}\cdots\frac{2n-1}{2n}\right)^2\cdots(5)$$

$$2^{\frac{1}{4}}=\prod_{n=1}^{\infty}\frac{e^{(2n-1)\pi}+1}{e^ {(3n-1)\pi}}\cdot\frac{(e^{n\pi}+1)^3}{e^{2n\pi}+1}\cdots(6)$$

$$\pi=\lim_{n\to\infty}\left[\prod_{k=1}^{n}\left(1+\frac{1}{2k-1}\right)^2 -\prod_{k=1}^{n-1}\left(1+\frac{1}{2k-1}\right)^2\right]\cdots(7)$$

$$\frac{e^{\pi}}{2^3}=\prod_{n=1,3,5,...}^{\infty}\left(\frac{e^{n\pi}} {e^{n\pi}-1}\right)^{24}\cdots(8)$$

$$\frac{e^{2\pi}}{2^9}=\prod_{n=1,3,5,...}^{\infty} \left(\frac{e^{2n\pi}}{e^{2n\pi}-1}\right)^{24}\cdots(9)$$

$$2=\prod_{n=1,3,5,...}^{\infty} \left(\frac{e^{n\pi}+1}{e^{n\pi}-1}\right)^{8}\cdots(10)$$

$$\left(\frac{2}{\pi}\right)^{\frac{1}{4}}\cdot{\Gamma\left(\frac{3}{4}\right)}=\prod_{n=1}^{\infty} \left(\frac{e^{n\pi}+1}{e^{n\pi}-1}\right)\cdots(11)$$

$$2=\prod_{n=1}^{\infty}\left[1+\frac{1}{e^{(2n-1)\pi}-1}\right]^{20}\left[\left(1+\frac{1} {e^{n\pi}}\right)\left(1-\frac{1}{e^{n\pi}}\right)^2\right]^4\cdots(12)$$

$$e^{\pi}=\prod_{n=1}^{\infty}\left(1+\frac{1}{e^{n\pi}-1}\right) ^{30}\left(1+\frac{1}{e^{(2n-1)\pi}-1}\right)^{42}\left(1-\frac{1}{e^{2n\pi}}\right)^{54}\cdots(13)$$

$$ln\left[\frac{\Gamma^2(\frac{1}{4})}{4\sqrt{2\pi}}\right]= \sum_{n=1}^{\infty}\frac{1-\beta(n)}{n}\cdots(1)$$

$$ln\left[\frac{\Gamma^{4}(\frac{1}{4})}{16\pi^2}\right]=\sum_{k=1}^{\infty} \frac{1-\beta(2k)}{k}\cdots(2)$$

$$ln\sqrt\frac{\pi}{2}=\sum_{k=0}^{\infty}\frac{1-\beta(2k+1)}{2k+1}\cdots(3)$$

$$ln\sqrt{\frac{7}{6}}=\sum_{k=0}^{\infty}\frac{1}{2k+1}\sum_{p=2}^ {\infty}\frac{1}{p^{8k+4}}\cdots(4)$$

$$ln\sqrt{\frac{5}{2}}=\sum_{k=0}^{\infty}\frac{1}{2k+1}\sum_{p=2}^{\infty}\frac{1} {p^{4k+2}}\cdots(5)$$

$$ln\left(\frac{\pi}{sinh\pi}\right)=\sum_{n=1}^{\infty}\frac{(-1) ^k\zeta(2n)}{n}\cdots(6)$$

$$ln\sqrt{\frac{3}{2}}=\sum_{k=0}^{\infty}\frac{\zeta(6k+3)-1}{2k+1}\cdots(7)$$

$$ln\sqrt{\frac{sinh\pi}{\pi}}=\sum_{k=0}^{\infty}\frac{\zeta(4k+2)-1}{2k+1}\cdots(8)$$

$$ln\pi-\gamma=\sum_{k=2}^{\infty}\frac{\zeta(k)}{k\cdot2^{k-1}}\cdots(9)$$

$$ln2-\gamma=\sum_{k=1}^{\infty}\frac{\zeta(2k+1)}{(2k+1)2^{2k}}\cdots(10)$$

$$ln(2\pi)=\sum_{n=1}^{\infty}\frac{\left(2^{2n+1}+1\right)\zeta(2n) -2^{2n+1}}{2^{2n}\cdot{n}}\cdots(11)$$

$$ln\left(\frac{\pi}{2}\right)=\sum_{n=1}^{\infty}\frac{2^{2n}n}\cdots(12)$$

$$ln\left(\frac{\pi}{3}\right)=\sum_{n=1}^{\infty}\frac{6^{2n}n}\cdots(13)$$

$$ln\left(\frac{\pi\phi}{5}\right)=\sum_{n=1}^{\infty} \frac{10^{2n}n}\cdots(14)$$

$$ln\left(\frac{2\pi\phi}{5\sqrt{\phi+2}}\right)= \sum_{n=1}^{\infty}\frac{5^{2n}n}\cdots(15)$$

$$ln(2)=\sum_{n=1}^{\infty}\frac{\zeta(2n)-1}{n}\cdots(16)$$

$$ln\left(\frac{4\pi}{sinh\pi}\right)=\sum_{n=1}^{\infty}\frac{\zeta(4n)-1}{n}\cdots(17)$$

$${\gamma}=\lim_{k\to\infty}\left[\sum_{n=1}^k{\frac{\zeta(2n)}{n}-ln2k}\right]\cdots(18)$$

$$ln\left(\frac{2}{\pi}\right)=\sum_{n=1}^{\infty}\frac{(-1)^n\eta(n)}{n}\cdots(19)$$

$$ln\sqrt2=\sum_{k=0}^{\infty}\frac{1-\eta(2k+1)}{2k+1}\cdots(20)$$

$$ln\left(\frac{8}{\pi^2}\right)=\sum_{n=1}^{\infty}\frac{\eta(2n)-1}{n}\cdots(21)$$

$$ln\left(\frac{2}{\pi}\right)=\sum_{n=1}^{\infty}\frac{\eta(n)-1}{n}\cdots(22)$$

$$ln\left(\frac{4}{\pi}\right)= \sum_{n=1}^{\infty}\frac{\lambda(2n)-1}{n}\cdots(23)$$

$$ln\left(\frac{1}{\pi}\right)+2-\gamma=2\sum_{n=2}^{\infty}\frac{\lambda(n)-1}{n}\cdots(24)$$

$${\gamma}=\lim_{k\to\infty}\left[\sum_{n=1}^k{\frac{\lambda(2n)}{m}-ln \left(\frac{4k}{\pi}\right)}\right]\cdots(25)$$

$$\gamma+ln\left(\frac{4}{\pi}\right)=2\sum_{n=2} ^{\infty}\frac{(-1)^n\lambda(n)}{n}\cdots(26)$$

$$\frac{\pi}{6}-\frac{3}{4}ln2=\sum_{k=1}^{\infty}\frac{1}{k}\left(\frac{1} {e^{2k\pi}+1}+\frac{1}{e^{2k\pi}-1}\right)\cdots(27)$$

$$\frac{1}{8}ln2=\sum_{k=0}^{\infty}\frac{1}{2k+1}\left (\frac{1}{e^{(2k+1)\pi}+1}+\frac{1}{e^{(2k+1)\pi}-1}\right)\cdots(28)$$

$$\frac{\pi}{12}-\frac{1}{2}ln2=\sum_{n=1}^{\infty}\frac{(-1)^n}{n}\left(\frac{1}{e^{n\pi}+1}+ \frac{1}{e^{n\pi}-1}\right)\cdots(29)$$

$$ln\frac{\pi^{\frac{1}{4}}}{\Gamma(\frac{3}{4})}=2\sum_{k=0}^{\infty} \frac{1}{(2k+1)(e^{\pi(2k+1)}+1)}\cdots(30)$$

$$ln\left[\left(\frac{2}{\pi}\right)^{\frac{1}{4}}\cdot{\Gamma\left(\frac{3}{4}\right)}\right]=2\sum_{k=0}^{\infty} \frac{1}{(2k+1)(e^{\pi(2k+1)}-1)}\cdots(31)$$

x > 1

$$ln2=\sum_{n=1}^{\infty}\frac{1}{(2^x-x-1)n}\sum_{r=1}^{2^x-2}\frac{2^x-r-1} {(r+1)^{2n}}\cdots(32)$$

$$ln3=\sum_{k=0}^{\infty}\frac{1}{2^{2k}\cdot(2k+1)}\cdots(33)$$

$$ln2=\sum_{n=1}^{\infty}\frac{1}{n}\left(\frac{1}{2^{2n}}+\frac{2}{3^{2n}} +\frac{2}{4^{2n}}+\frac{1}{5^{2n}}\right)\cdots(34)$$

$$ln\left(\frac{2\sqrt2}{e}\right)=\sum_{n=1}^{\infty}\frac{1}{n}\left[\frac{1}{4}\left(\frac{1}{3^{2n}}\right)+ \frac{1}{8}\left(\frac{1}{5^{2n}}+\frac{1}{7^{2n}}\right)+\frac{1}{16}\left(\frac{1}{9^{2n}}+\frac{1}{11^{2n}}+ \frac{1}{13^{2n}}+\frac{1}{15^{2n}}\right)+\cdots\right]\cdots(35)$$

$$\sum_{k=1}^{\infty}\frac{\zeta(2k)\left[(2^{2k}-1)\zeta(2k)-2^{2k}\right]}{k}+ln\left(\frac{\pi}{2}\right)= \sum_{n=1}^{\infty}\frac{1}{n}\sum_{r=1}^{\infty}\sum_{a=3}^{\infty}\left[\frac{1}{(2ar-a-1)^{2n}} +\frac{2}{(2ar-a)^{2n}}+\frac{1}{(2ar-a+1)^{2n}}\right]\cdots(36)$$

'''Where K = 8.700... is the constant from polygon circumscribing.'''

$$ln\left(\frac{2K}{\pi}\right)=\sum_{n=1}^{\infty}\frac{1}{n}\sum_{r=1}^{\infty} \sum_{k=1}^{\infty}\left[\frac{1}{(2rk+r-1)^{2n}}+\frac{2}{(2rk+r)^{2n}} +\frac{1}{(2rk+r+1)^{2n}}\right]\cdots(37)$$

Infinite sum

$$\pi=60\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n(e^{n\pi}-1)}+24\sum_{n=1}^{\infty} \frac{(-1)^n}{n(e^{2n\pi}-1)}+6\sum_{n=1}^{\infty}\frac{1}{n}\left(\frac{1} {e^{n\pi}-1}+\frac{1}{e^{n\pi}+1}\right)\cdots(1)$$