Well it is relatively well known that the condition for absolute convergence is given by the following theorem: In order that the infinite product $\prod _{n=1}^{\infty }\left( 1+a_{n}\right) $ may be absolutely convergent, it is necessary and sufficient that the series $\sum _{n=1}^{\infty }a_{n}$ should be absolutely convergent.
I am trying to prove a little less famous result from Cauchy, which states If $\sum _{n=1}^{\infty }a_{n}$ be a conditionally convergent series of real terms, then $\prod _{n=1}^{\infty }\left( 1+a_{n}\right) $ converges (but not absolutely) or diverges to zero according as $\sum _{n=1}^{\infty }a_{n}^{2}$ converges or diverges.
Some thoughts towards the Proof Although i could eb wrong here but since we do not know that $a_{n}\rightarrow 0$ under the given circumstances i guess a proof by comparison to $\sum _{k=0}^{\infty }\dfrac {1} {k^{2}}$ or which required the ln series kind of seem to fall apart. I was hoping some one could possibly provide an idea/ strategy for this proof.
Any help would be much appreciated.