Analytic Constant Microscopy

Say I have a constant, pi, or e, or what have you which: A. occurs in infinite series/products,  and B. infinite series/products representations for it exist

Ramanujan’s:

$latex \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum^{\infty}_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}$

$\pi_{n} = \left[\frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum^{n}_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\right]^{-1}$

If the Dirichlet Eta function is

$latex \eta(\tau) = e^{\frac{\pi \rm{i} \tau}{12}} \prod_{n=1}^{\infty} (1-(e^{2\pi i \tau})^{n})$

Then a Ramanujan-pi-formula-approximant-deformed Dirichlet Eta function is:

$latex \eta(\tau) = e^{\frac{\pi \rm{i} \tau}{12}} \prod_{n=1}^{\infty} (1-(e^{2\pi_{n} i \tau})^{n})$

with $\pi_{n}$ as defined above.