I want to use $S = \sum_{k=0} ^\infty (-1)^k \frac{(2k)!!}{(2k+1)!!} a^{2k+1}$ and get the relation $(a^2+1) S'=1−aS$. So far I am just getting $\frac{dS}{da} = \sum_{k=0} ^\infty (-1)^k \frac{(2k)!!}{(2k+1)!!} a^{2k} (2k+1)$, which I am not seeing how to use with S to get that relation. What should I do to progress and get the desired relation?
2026-03-01 05:19:04.1772342344
$\sum_{k=0} ^\infty (-1)^k \frac{(2k)!!}{(2k+1)!!} a^{2k+1}$ derivative relation
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