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Fungrim entry: da1873

n=01F2n+1=54θ22 ⁣(0,τ)   where τ=1πilog ⁣(352)\sum_{n=0}^{\infty} \frac{1}{F_{2 n + 1}} = \frac{\sqrt{5}}{4} \theta_{2}^{2}\!\left(0, \tau\right)\; \text{ where } \tau = \frac{1}{\pi i} \log\!\left(\frac{3 - \sqrt{5}}{2}\right)
References:
  • J. M. Borwein and P. B. Borwein. Pi and the AGM. Wiley, New York, 1987.
TeX:
\sum_{n=0}^{\infty} \frac{1}{F_{2 n + 1}} = \frac{\sqrt{5}}{4} \theta_{2}^{2}\!\left(0, \tau\right)\; \text{ where } \tau = \frac{1}{\pi i} \log\!\left(\frac{3 - \sqrt{5}}{2}\right)
Definitions:
Fungrim symbol Notation Short description
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
FibonacciFnF_{n} Fibonacci number
Infinity\infty Positive infinity
Sqrtz\sqrt{z} Principal square root
Powab{a}^{b} Power
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
ConstPiπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
Loglog ⁣(z)\log\!\left(z\right) Natural logarithm
Source code for this entry:
Entry(ID("da1873"),
    Formula(Equal(Sum(Div(1, Fibonacci(Add(Mul(2, n), 1))), Tuple(n, 0, Infinity)), Where(Mul(Div(Sqrt(5), 4), Pow(JacobiTheta(2, 0, tau), 2)), Equal(tau, Mul(Div(1, Mul(ConstPi, ConstI)), Log(Div(Sub(3, Sqrt(5)), 2))))))),
    References("J. M. Borwein and P. B. Borwein. Pi and the AGM. Wiley, New York, 1987."))

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2019-09-20 18:07:53.062439 UTC