# Fungrim entry: da1873

$\sum_{n=0}^{\infty} \frac{1}{F_{2 n + 1}} = \frac{\sqrt{5}}{4} \theta_{2}^{2}\!\left(0, \tau\right)\; \text{ where } \tau = \frac{1}{\pi i} \log\!\left(\frac{3 - \sqrt{5}}{2}\right)$
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
Sum$\sum_{n} f(n)$ Sum
Fibonacci$F_{n}$ Fibonacci number
Infinity$\infty$ Positive infinity
Sqrt$\sqrt{z}$ Principal square root
Pow${a}^{b}$ Power
JacobiTheta$\theta_{j}\!\left(z , \tau\right)$ Jacobi theta function
Pi$\pi$ The constant pi (3.14...)
ConstI$i$ Imaginary unit
Log$\log(z)$ Natural logarithm
Source code for this entry:
Entry(ID("da1873"),
Formula(Equal(Sum(Div(1, Fibonacci(Add(Mul(2, n), 1))), For(n, 0, Infinity)), Where(Mul(Div(Sqrt(5), 4), Pow(JacobiTheta(2, 0, tau), 2)), Equal(tau, Mul(Div(1, Mul(Pi, ConstI)), Log(Div(Sub(3, Sqrt(5)), 2))))))),
References("J. M. Borwein and P. B. Borwein. Pi and the AGM. Wiley, New York, 1987."))

## Topics using this entry

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2020-08-27 09:56:25.682319 UTC