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Fungrim entry: 22b67a

n=11Fn2=524(θ24 ⁣(0,τ)θ44 ⁣(0,τ)+1)   where τ=1πilog ⁣(352)\sum_{n=1}^{\infty} \frac{1}{F_{n}^{2}} = \frac{5}{24} \left(\theta_{2}^{4}\!\left(0, \tau\right) - \theta_{4}^{4}\!\left(0, \tau\right) + 1\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=1}^{\infty} \frac{1}{F_{n}^{2}} = \frac{5}{24} \left(\theta_{2}^{4}\!\left(0, \tau\right) - \theta_{4}^{4}\!\left(0, \tau\right) + 1\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
Powab{a}^{b} Power
FibonacciFnF_{n} Fibonacci number
Infinity\infty Positive infinity
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
Sqrtz\sqrt{z} Principal square root
Source code for this entry:
Entry(ID("22b67a"),
    Formula(Equal(Sum(Div(1, Pow(Fibonacci(n), 2)), For(n, 1, Infinity)), Where(Mul(Div(5, 24), Add(Sub(Pow(JacobiTheta(2, 0, tau), 4), Pow(JacobiTheta(4, 0, tau), 4)), 1)), 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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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-09-22 15:43:45.410764 UTC