Answer by r9m for How to prove this infinite product identity?
The identity is in fact true for any $0 < x < 1$: $$\prod_{k=1}^{\infty}\frac{1}{1-x^{2k-1}}=\sum_{m=0}^{\infty}\left(x^{\frac{m^2+m}{2}}\prod_{n=1}^{\infty}\frac{1-x^{m+n}}{1-x^{n}}\right) =...
View ArticleHow to prove this infinite product identity?
How can I prove the following identity?$$\large\prod_{k=1}^\infty\frac1{1-2^{1-2k}}=\sum_{m=0}^\infty\left(2^{-\frac{m^2+m}{2}}\prod_{n=1}^\infty\frac{1-2^{-m-n}}{1-2^{-n}}\right)$$Numerically both...
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