Finite and infinite expansion of nested Radical involving Golden ratio as follows
$2\cos\frac{2\pi}{5} = \frac{1}{\phi} $ where $\phi$ is Golden ratio $(2\cos72°)$
Let us expand this with increasing nested square roots
$\sqrt{2-2\cos\frac{2\pi}{5}} =\sqrt{2+2\cos\frac{3\pi}{5}} =2\cos\frac{3\pi}{2\cdot5} = \sqrt{2- \frac{1}{\phi}} =\sqrt{2-(\phi-1)} =\sqrt{3-\phi}$ $(2\cos54°)$
$\sqrt{2-\sqrt{2-2\cos\frac{2\pi}{5}}}=\sqrt{2-\sqrt{3-\phi}} = \sqrt{2-2\cos\frac{3\pi}{2\cdot5}} =2\cos\frac{7\pi}{2^2\cdot5}= (2\cos63°)$
$\sqrt{2-\sqrt{2-\sqrt{2-2\cos\frac{2\pi}{5}}}}=\sqrt{2-\sqrt{2-\sqrt{3-\phi}}} = \sqrt{2-2\cos\frac{7\pi}{2^2\cdot5}} =2\cos\frac{13\pi}{2^3\cdot5}= (2\cos58.5°)$
$\sqrt{2-\sqrt{2-\sqrt{2-\sqrt{2-2\cos\frac{2\pi}{5}}}}}=\sqrt{2-\sqrt{2-\sqrt{2-\sqrt{3-\phi}}}} = \sqrt{2-2\cos\frac{13\pi}{2^3\cdot5}} =2\cos\frac{27\pi}{2^4\cdot5}= (2\cos60.75°)$
Subsequent angles with increasing nested radical with $-$ signs as follows
$2\cos\frac{53\pi}{2^5\cdot5}=(2\cos59.625°)$
$2\cos\frac{107\pi}{2^6\cdot5}=(2\cos60.1875°)$
$2\cos\frac{213\pi}{2^5\cdot5}=(2\cos59.90625°)$ and so on
The number series in the numerator is as follows
$2,3,7,13,27,53,107,213,... $refer here $a(n) = a(n-1) + 2*a(n-2), a(0)=2, a(1)=3.$
And can be derived also as follows $\frac{(2^n\times5)+(-1)^n}{3}$ where n refers number of $(-)$ signs in the Nested radical.
1st question:
Can anyone explain signed bits and fast exponentiation associated with this number series
Oscillation of angle on either side of $\frac{\pi}{3}$ (just like a pendulum)
Angles derived as follows
$60^\circ + 12^\circ = 72^\circ$
$60^\circ - 6^\circ = 54^\circ$
$60^\circ + 3^\circ = 63^\circ$
$60^\circ - 1.5^\circ = 58.5^\circ$
$60^\circ + 0.75^\circ = 60.75^\circ$
$60^\circ - 0.375^\circ = 59.625^\circ$
$60^\circ + 0.1875^\circ = 60.1875^\circ$
$60^\circ - 0.09275^\circ = 59.90625^\circ$ . . .
2nd question
Is there an anyway to link calculus with increasing nested radical with application of limits?