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See page 5 in http://arxiv.org/abs/1112.3259 Are the series presented there somewhat related to your series (7)?

Using (see entry 24 on page 291 in Ramanujan's Notebooks II: http://www.plou) $$\frac{\pi e^{-2\pi z}}{2z[\cosh{(2\pi z)}-\cos{(2\pi z)}]}= \frac{1}{8\pi z^3}-\frac{1}{4z^2}+\frac{\pi}{4z}-\sum\limits …

As it is well known, Ramanujan's $\tau(n)$ function can be defined through the power series expansion of the modular discriminant: $$\Delta(q)=q\prod\limits_{n=1}^\infty (1-q^n)^{24}=\sum \limits_{n …

The result $$\small R(e^{-2\pi\sqrt{5}})=\frac{\sqrt{5}}{1+\left[5^{3/4}\left(\frac{\sqrt{5}-1}{2}\right)^{5/2}-1\right]^{1/5}}-\frac{\sqrt{5}+1}{2}=\frac{\beta+2}{\beta+\sqrt[5]{\sqrt{1+\beta^{10}}-\ …

A general methods of construction of Ramanujan-type identities are outlined in http://arxiv.org/abs/1211.6563 (Some conjectured formulas for 1/Pi coming from polytopes, K3-surfaces and Moonshine, by G …

Try this reference https://arxiv.org/abs/1105.5759 (Notes on "Quadratic Forms and Automorphic Forms" from the 2009 Arizona Winter School, by J. Hanke). P.S. See also http://citeseerx.ist.psu.edu/view …