The beauty of Infinity


Infinity and its infinite beauty.



Today we're going to take a brief look at the beauty of infinity, through:


to represent some of the most well known and beautiful mathematical constants, like π, e and φ.

# Definitions

Before diving into it, let's begin with some simple definitions:

1) Infinite sum:

n=11n=11 +12 +13 +...

2) Infinite product:

n=11n=11 12 13  ...


3) Continued fractions:

Kn=11n=11+12 +13+...

4) Nested radicals:

Rn=1(n +R)12=1+ 2+ 3+ ...

5) Limit:

limn(1+1n)n=(1+1n)n, where n approaches .

# Infinite sums


n=01n!=e

n=0xnn!=ex

n=0nxn!=eBellx

n=0x(-1)nn!=xe

n=0n!(2n+1)!!=π2

n=0n!(2n)!!(2n+1)!=π2

n=0n!(2n+2)!!=log(2)

n=01(2n+2)2n=log(2)

n=01(2n+1)(2n+2)=log(2)

n=1(2n-1)!22n(n!)2=log(2)

n=01(2n+1)2n=2log(1+2)

n=0(2n+1)!!(2n+1)!=e

n=0(n!)2(2n+1)!=2π33

n=02n+1(n!)2(2n+1)!=π

n=0(-1)n2n+1=π4

n=0i2n+12n+1=πi4

n=0(i2-i)2n+12n+1 =πi8

n=11n2=π26

n=11n(n+1)2=2-π26

n=0(-1)n(2n+1)3=π332

n=01n+ (2n+1)2=π6+ln(2)

n=01(2n+1)2=π28

n=0(-1)n(n+1)2=π212

n=022nB2n(2n)!=e2+1e2-1

n=0(2x)2nB2n(2n)!=xcoth(x),      0<|x|<π.

n=0(-1)n(2x)2nB2n(2n)!=xcot(x),      0<|x|<π.

n=122n(22n-1)x2n-1B2n(2n)!=tanh(x),      |x|<π2.

n=1(-1)n-122n(22n-1)x2n-1B2n(2n)!=tan(x),      |x|<π2.

n=01(2n+x)k=-ψ(k-1)(x2)2k(k-1)!

n=0(-1)n(2n+1)2=β(2)

n=11n+n2=1

n=0x2n(-1)n(2n)!=cos(x)

n=0x2n+1(-1)n(2n+1)!=sin(x)

n=0(xi)2n+1-(-xi)2n+1(2n+1)!=2csc(x)i

n=0x2n(2n)!=cosh(x)

n=0x2n+1(2n+1)!=sinh(x)

n=0(2x)nn!=21-tanh(x)-1

n=0(2x)nn!=1-21-coth(x)

n=0x2n+1-(-x)2n+1(2n+1)!=2csch(x)

n=0(x2+1-1xi)2n+12n+1 =tan-1(x)2i

n=0(x2+1-1x)2n+12n+1 =sinh-1(x)2

n=0(1-1-x2xi)2n+12n+1 =sin-1(x)2i

n=0(1-1-x2x)2n+12n+1 =tanh-1(x)2

n=0(x-1x+1i)2n+12n+1 =sec-1(x)2i

n=0x-1x+12n+12n+1 =cosh-1(x)2

n=0(1-x1+xi)2n+12n+1 =cos-1(x)2i

n=01-x1+x2n+12n+1 =sech-1(x)2

n=0((1-1-1x2)xi)2n+12n+1 =csc-1(x)2i

n=0((1-1-1x2)x)2n+12n+1 =coth-1(x)2

n=0((1x2+1-1)xi)2n+12n+1 =cot-1(x)2i

n=0((1x2+1-1)x)2n+12n+1 =csch-1(x)2

n=0x-2n-12n+1=coth-1(x)    for |x| > 1

n=0x2ny(2n)!=cosh(x1y)

n=0x2n+1y(2n+1)!=sinh(x1y)

n=01n2 +(n+1)2=tanh(π2)π2

n=0(-1)nn+1=ln(2)

n=0inn+1=π4+log(2)i

n=0inn!=cos(1)+sin(1)i

n=0in(2n)!=cosh((-1)14)

n=11nkkn=Lik(1k)    for |k| > 1

n=0inen=ee-i

n=0(x-1x+1)2n+12n+1 =ln(x)2

n=0mk=2(k-1k+1)2n+12n+1=ln(m!)2

n=0jk=0(ak)nn!=exp(jk=0ak)

n=0(-14)ncos-1(3)2n(2n)!=2

n=0ζ(2n)(2n+1)(2n+2)22n=-14ζ(3)8π2

[log(2716)+n=0ζ(2n)-1(2n+1)(2n+2)22n]-8π214=ζ(3)

n=0(-14)n(11+2n +21+4n +13+4n)=π

n=1log(n)n2=-π26(γ-12log(A)+log(2π))


# Infinite products


p11-p-2= π26

n=14n24n2-1=π2

n=0exp(1n!)=ee

n=14n2+14n2-1=sinh(π2)

n=14n2-14n2+1=csch(π2)

n=14n34n3-3n+1=π

n=1(2n+1)2-1(2n+1)2=π4

n=1(1-14n2)=2π

n=1(1-x2π2n2)=sin(x)x

n=0(1-4x2π2(2n+1)2)=cos(x)

n=1xn2xn2-1=πxsin(πx)

n=1xπ2n2xπ2n2-1=csc(1x)x

n=1(1 +1n)x(1 +xn)-1 =Γ(x+1)

\prod_{n=1}^(\infty)\exp(\frac{1}{n}-2) (\frac{1}{n} + 1)^(2n) = \frac{\exp(2+\gamma)}{2\pi}

# Continued fractions


\mathbf{K}(\frac{1}{1}) = φ - 1

\mathbf{K}(\frac{1}{2}) = \sqrt(2) - 1

\mathbf{K}(\frac{x-1}{2}) = \sqrt(x) - 1

\mathbf{K}(\frac{1}{x}) = \frac{\sqrt(x^2+4) - x}{2}

\mathbf{K}(\frac{x}{1}) = \frac{\sqrt(4x+1) - 1}{2}

\mathbf{K}(\frac{x}{y}) = \frac{\sqrt(y^2 + 4x) - y}{2}

\underset{n=0}{\overset{\infty}{\mathbf{K}}}(\frac{(-1)^n}{1}) = φ

\underset{n=1}{\overset{\infty}{\mathbf{K}}}(\frac{n}{n}) = \frac{1}{\e-1}

\underset{n=1}{\overset{\infty}{\mathbf{K}}}(\frac{1}{2n + 1}) = \frac{2}{\e^2-1}

\underset{n=0}{\overset{\infty}{\mathbf{K}}}(\frac{1}{2n + 1}) = \tanh(1)

\underset{n=1}{\overset{\infty}{\mathbf{K}}}(\frac{2n}{n-1}) = \coth(1)

\underset{n=0}{\overset{\infty}{\mathbf{K}}}(\frac{(2n+1)^2}{6}) = \pi-3

\underset{n=0}{\overset{\infty}{\mathbf{K}}}(\frac{(2n+1)^2}{2}) = \frac{4}{\pi}-1

\underset{n=1}{\overset{\infty}{\mathbf{K}}}(\frac{n^2}{2n + 1}) = \frac{4}{\pi}-1

\underset{n=1}{\overset{\infty}{\mathbf{K}}}(\frac{n^4}{2n + 1}) = \frac{12}{\pi^2}-1

\frac{2}{1+\underset{n=1}{\overset{\infty}{\mathbf{K}}}(\frac{n^4}{2n+1})} = \zeta(2)

1+\frac{1}{5+\underset{n=1}{\overset{\infty}{\mathbf{K}}}(\frac{-n^6}{(2n+1)*(n^2+n+5)})} = \zeta(3)

\underset{n=1}{\overset{\infty}{\mathbf{K}}}(\frac{\floor(\frac{n+1}{2}) * \floor(\frac{n+2}{2})}{1}) = \frac{6}{\pi^2 - 6}-1

1+\frac{x}{1 + \underset{n=1}{\overset{\infty}{\mathbf{K}}}(\frac{-n * x}{x + n + 1})} = \exp(x)

\frac{x}{1+\underset{n=1}{\overset{\infty}{\mathbf{K}}}(\frac{n^2 * x^2}{2n+1})} = \tan^-1(x)

\frac{1}{1+\underset{n=1}{\overset{\infty}{\mathbf{K}}}(\frac{\frac{x^2}{4n^2 - 2n}}{1-\frac{x^2}{4n^2 - 2n}})} = \cos(x)

\frac{x}{1+\underset{n=1}{\overset{\infty}{\mathbf{K}}}(\frac{\frac{x^2}{4n^2 + 2n}}{1-\frac{x^2}{4n^2 + 2n}})} = \sin(x)


# Nested radicals

\prod_{n=1}^(\infty)\frac{2}{\underset{1}{\overset{n}{\mathbf{R}}}(2+R)^(1/2)} = \frac{\pi}{2}

(1+R)^(1/2) = \sqrt(1+\sqrt(1+\sqrt(1+\...))) = φ

(x+R)^(1/2) = \sqrt(x+\sqrt(x+\sqrt(x+\...))) = \frac{1 + \sqrt(4x+1)}{2}

(1+R)^(1/3) = (1 + (1 + (1 + \...)^(1/3))^(1/3))^(1/3) = \rho

\lim_{n to \infty}((2 - \underset{1}{\overset{n}{\mathbf{R}}}(2 + R)^(1/2)) * 4^(n+1)) = (2 - \sqrt(2 + \sqrt(2 + \sqrt(2 + \...)))) * 4^(n+1) = \pi^2

\underset{n=1}{\overset{\infty}{\mathbf{R}}}(1 + n * R)^(1/2) = \sqrt(1 + 1*\sqrt(1 + 2*\sqrt(1 + 3*\...))) = 2

\underset{n=x}{\overset{\infty}{\mathbf{R}}}(1 + n * R)^(1/2) = \sqrt(1 + x*\sqrt(1 + (x+1)*\sqrt(1 + (x+2)*\...))) = x+1

# Limits

\lim_{n to \infty}(1 + 1/n)^n = \e

\lim_{n to \infty}(\frac{n-1}{n})^(-n) = \e

\lim_{n to \infty}[prod_{p}^(n) p^(1/n)] = \e

\lim_{n to \infty}[prod_{k=1}^(n) p_k^(1/(n \ln n))] = \e

\lim_{n to \infty}[\H_n - \ln n] = \gamma

\lim_{n to \infty}\frac{\exp(\H_n)}{n} = \exp(\gamma)

\lim_{n to \infty}[\zeta((n+1)/n) - n] = \gamma

\lim_{n to \infty}\frac{(n!)^2 * e^(2n)}{n^(2n+1)} = 2\pi

\lim_{n to \infty}\frac{n!}{\sqrt(n) * e^(-n) * n^n} = \sqrt(2\pi)

\lim_{n to \infty}[sum_{k=0}^(n)\frac{n! * n^(k-n)}{k!*(n-k)!}] = \e

\lim_{n to \infty}[(n-1)^(-n) * (n+1)^n] = \e^2

\lim_{n to \infty}[n^(1-n) * (n + x)^(n-1)] = \exp(\x)

lim_{n to \infty}\frac{2^(4n)*n*(n!)^4}{((2n+1)!)^2} = \frac{\pi}{4}

lim_{n to \infty}\frac{2^(4n-1) * (n!)^4}{(2n+1) * ((2n)!)^2} = \frac{\pi}{4}

lim_{n to \infty}\frac{2^(4n)*(n!)^3*(n-1)!}{((2n)!)^2} = \pi

lim_{n to \infty}\frac{2^(4n)*(n!)^3*(n+1)!}{((2n+1)!)^2} = \frac{\pi}{4}

lim_{n to \infty}\frac{4^n * (n!)^2}{(2n-1)!! * (2n+1)!!} = \frac{\pi}{2}

lim_{n to \infty}\frac{2^(8n)*n*(n+1)!*(n!)^7}{((2n+1)!)^4} = \frac{\pi^2}{16}

lim_{n to \infty}\frac{2^(8n+1)*(n+1)^4*(2n^2 + 2n + 1) * (n!)^8}{((2n+2)!)^4} = \frac{\pi^2}{8}

lim_{n to \infty}[n^2/4 - \frac{6n^2+6n+1}{12}*log(n) + sum_{k=1}^(n)k \ln k] = \log(\A)

# Asymptotics

sqrt(2\pi\n) * (n/e)^n ~~ n!

2^n * sqrt(2 \pi n) * (n/e)^n ~~ (2n)!!

2^(n+1/2) * (n/e)^n ~~ (2n-1)!!

2^(n+3/2) * ((n+1)/e)^(n+1) ~~ (2n+1)!!

n * (log(n) + log(2) - 1) + log(2 \pi n)/2 ~~ log((2n)!!)

n*log(n) + (n + 1/2)*log(2) - n ~~ log((2n-1)!!)

(n+1)*log(n+1) + (n + 3/2)*log(2) - n - 1 ~~ log((2n+1)!!)

\frac{2}{(2\pi)^n} * n! ~~ |B_n|

\sqrt(8\pi\n) * (\frac{n}{2\pi\e})^n ~~ |B_n|

\log(\sqrt(8\pi\n)) + n * (log(n) - log(2\pi) - 1) ~~ log(|B_n|)

-\frac{n^2}{4} + \frac{6n^2 + 6n + 1}{12} * log(n) + log(\A) ~~ \sum_{k=1}^(n) k \ln k

\sqrt(2\pi\n) * \exp(n*log(n) - n) * (\log(n) + \gamma) ~~ n!*\sum_{k=1}^(n) \frac{1}{k}

\log(\sqrt(2\pi\n)) + n*log(n) + log(log(n) + \gamma) - n ~~ log(n!*\sum_{k=1}^(n) \frac{1}{k})

\sqrt(2\pi) * (\frac{12n^2 + 24n + 12 + 1}{12\e\n + 12\e})^(n+1) * \frac{1}{\sqrt(n+1)} ~~ n!

1/2 * (\log(\frac{2\pi}{n+1}) - 2*(n+1) * \log(\frac{12\e\n + 12\e}{12n^2+24n+12+1})) ~~ \log(n!)

1/6 * (\log(8n^3 + 4n^2 + n + 1/30) - 6n + 6n*log(n) + 3\log(\pi)) ~~ \log(n!)

lim_{k to \infty}[\sqrt(2\pi) * (\frac{12n^2 + 24kn + 12k^2 + 1}{12\e\n + 12\e\k})^(n+k) * \frac{n!}{(n+k-1)!} * \frac{1}{\sqrt(n+k)}] ~~ n!

# Implementations:

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