New posts in golden-ratio

What is this function related with continued fractions?

real-analysis sequences-and-series recurrence-relations continued-fractions golden-ratio

Prove $_2F_1\!\left(\frac76,\frac12;\,\frac13;\,-\phi^2\right)=0$

calculus special-functions hypergeometric-function golden-ratio

Fibonacci numbers and golden ratio: $\Phi = \lim \sqrt[n]{F_n}$

fibonacci-numbers golden-ratio

A series with Fibonacci numbers and the golden ratio

calculus sequences-and-series trigonometry fibonacci-numbers golden-ratio

What's the value of $n+\frac{n}{n+\frac{n}{n+\frac{n}{\ddots}}}$ for $n\in\mathbb{C}$?

number-theory continued-fractions golden-ratio

A pattern appearing in the powers of $\phi$

number-theory golden-ratio

$\int_{0}^{\pi/2}\ln\left(1+4\sin^4 x\right)\mathrm{d}x$ and the golden ratio

calculus integration definite-integrals logarithms golden-ratio

Prove that $\int _{-\infty }^{+\infty }{\frac {\mathrm {d} z}{(\phi ^{n}z)^{2}+(F_{2n+1}-\phi F_{2n})(e^{\gamma }z^{2}+\zeta (3)z-\pi )^2}}=1$

improper-integrals pi golden-ratio euler-mascheroni-constant

$\int_{0}^{1}{(1-x)(1-2x^{\phi})+\phi(x-x^{\phi})\over (1-x)^2}\cdot{\left(1-x^{\phi}\over 1-x\right)^{1\over \phi}}\mathrm dx=\phi^{\phi}$

calculus integration definite-integrals golden-ratio

New very simple golden ratio construction incorporating a triangle, square, and pentagon all with sides of equal length. Is there any prior art?

geometry trigonometry euclidean-geometry golden-ratio

Dodecahedron and golden ratio algebra

algebra-precalculus polyhedra golden-ratio

Does $\sum\limits_{n=1}^{\infty}\frac{1}{P_n\ln(P_n)}$ converge to the golden ratio?

sequences-and-series number-theory reference-request prime-numbers golden-ratio

Limit of the ratio of consecutive Fibonacci numbers [duplicate]

calculus sequences-and-series limits fibonacci-numbers golden-ratio

Charming approximation of $\pi$: $2\left(\frac{1}{2}\right)^{\phi/2}+2< \pi$, where $\phi$ is the golden ratio

inequality approximation alternative-proof pi golden-ratio

pandigital rational approximations to the golden ratio and the base of the natural logarithm

algebra-precalculus algorithms recreational-mathematics number-systems golden-ratio

How to prove $_2F_1\big(\tfrac16,\tfrac16;\tfrac23;-2^7\phi^9\big)=\large \frac{3}{5^{5/6}}\,\phi^{-1}\,$ with golden ratio $\phi$?

calculus definite-integrals hypergeometric-function conjectures golden-ratio

Prove that $\sum_{n=1}^\infty \left(\phi-\frac{F_{n+1}}{F_{n}}\right)=\frac{1}{\pi}$

limits summation golden-ratio

The most complex formula for the golden ratio $\varphi$ that I have ever seen. How was it achieved?

exponential-function pi continued-fractions golden-ratio ramanujan-summation

Uses of the 'Golden Ratio'

soft-question math-history golden-ratio

Another way to go about proving the limit of Fibonacci's sequence quotient.

sequences-and-series inequality fibonacci-numbers golden-ratio