Suppose I have a convergent infinite series $\sum_{n=0}^\infty (-1)^n a_n = S_0$ and $0 < S_0 < 1$. Write $s_n$ for the $n$-th partial sum. ($s_n = \sum_{k=0}^n (-1)^k a_k$) Now consider the new series $$S_1 = \sum_{n=0}^\infty (-1)^n \left ( \prod_{k = 1}^n s_k \right)$$ where we understand the empty product to be 1 (so that this is the first term.) This new series converges by the ratio test.

Can anything be said about its value? For example, if we always have $0 < S_1 < 1$, we could iterate the transformation to get a sequence of values $S_0, S_1, S_2, \dots$. Can anything be said about this sequence? I guess I'm wondering if anyone has seen this type of thing before (maybe it has a name?)