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!=e⋅Bellx
∞∑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=√2⋅log(1+√2)
∞∑n=0(2n+1)!!(2n+1)!=√e
∞∑n=0(n!)2(2n+1)!=2π3√3
∞∑n=02n+1⋅(n!)2(2n+1)!=π
∞∑n=0(-1)n2n+1=π4
∞∑n=0i2n+12n+1=πi4
∞∑n=0(i√2-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=022n⋅B2n(2n)!=e2+1e2-1
∞∑n=0(2x)2n⋅B2n(2n)!=x⋅coth(x), 0<|x|<π.
∞∑n=0(-1)n⋅(2x)2n⋅B2n(2n)!=x⋅cot(x), 0<|x|<π.
∞∑n=122n⋅(22n-1)⋅x2n-1⋅B2n(2n)!=tanh(x), |x|<π2.
∞∑n=1(-1)n-1⋅22n⋅(22n-1)⋅x2n-1⋅B2n(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=0√x-1x+12n+12n+1 =cosh-1(x)2
∞∑n=0(√1-x1+xi)2n+12n+1 =cos-1(x)2i
∞∑n=0√1-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=11nk⋅kn=Lik(1k) for |k| > 1
∞∑n=0inen=ee-i
∞∑n=0(x-1x+1)2n+12n+1 =ln(x)2
∞∑n=0∑mk=2(k-1k+1)2n+12n+1=ln(m!)2
∞∑n=0∏jk=0(ak)nn!=exp(j∏k=0ak)
∞∑n=0(-14)n⋅cos-1(3)2n(2n)!=√2
∞∑n=0ζ(2n)(2n+1)⋅(2n+2)⋅22n=-14⋅ζ(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-12⋅log(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-x2π2⋅n2)=sin(x)x
∞∏n=0(1-4x2π2⋅(2n+1)2)=cos(x)
∞∏n=1x⋅n2x⋅n2-1=π√x⋅sin(π√x)
∞∏n=1x⋅π2⋅n2x⋅π2⋅n2-1=csc(1√x)√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!
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