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?)

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    $\begingroup$ Some motivation would be helpful. This is a strange construction. $\endgroup$ – Qiaochu Yuan Aug 8 '10 at 21:46
  • $\begingroup$ Yes, this is rather peculiar. My reading of the literature of series transformation has never encountered multiplying together partial sums. $\endgroup$ – J. M. is not a mathematician Aug 9 '10 at 1:35
  • $\begingroup$ I am a bit lazy to calculate : look at what happens with $a_k=(1/2)^k$, and more generally $a_k=(1/p)^k$ can you calculate the $S_i$ $\endgroup$ – Jérôme JEAN-CHARLES Oct 8 '10 at 1:29

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