# deforming geometric series

Yesterday I asked if there could be such a sequence of real numbers $\{a_{n}\}_{\infty}$ such that for all $n\in\mathbb{N}$, AGM$(a_{n},a_{n+1}) = 1/2^{n}$, and it turns out that the new sequence is just a linearly stretched version, and that had me wondering, in what cases are we aware of nonlinearly stretched geometric sequences:

The ordinary geometric sequence:
$\sum_{n=1}^{\infty} \frac{1}{2^{n}} = 1$

The exponential:
$\sum_{n=1}^{\infty} \frac{1}{2^{n}n!} = \sqrt{e}-1$

“Wahi” or “Upsilon” functions:
$\sum_{n=1}^{\infty} \frac{1}{2^{n}n^{n}} = \int_{0}^{1} \frac{dx}{2\sqrt{x^{x}}} = \Upsilon_{0}(1/2) =$ʍ$(1/2)-1$

The polylogarithm (in particular, the dilogarithm):
$\sum_{n=1}^{\infty} \frac{1}{2^{n}n^{2}} = Li_2(1/2)$

But there are lacunary monsters like the following:
$\sum_{n=1}^{\infty} \frac{1}{2^{n}3^{2^{n}}}$
That there are no present methods of evaluation for.