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I am wondering whether the following two formal power series are equal: $A(x)=\Pi_{k=1}^{\infty}\frac{1}{1-x^{2k-1}}$, $B(x)=\Pi_{k=1}^{\infty}(1+x^k)$.

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  • $\begingroup$ A and B are not series. $\endgroup$ – William Elliot Mar 4 '18 at 8:32
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    $\begingroup$ @WilliamElliot: They are both series: $A(x)=\sum_{n=0}^\infty p(n|\text{parts all odd})x^n$ and $B(x)=\sum_{n=0}^\infty p(n|\text{parts distinct})x^n$. Here $p(n)$ is the number of partitions of $n$. $\endgroup$ – Markus Scheuer Mar 4 '18 at 8:37
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We obtain \begin{align*} \color{blue}{\prod_{k=1}^\infty(1+x^k)}&=\prod_{k=1}^\infty\frac{(1+x^k)(1-x^k)}{1-x^k}=\prod_{k=1}^\infty\frac{1-x^{2k}}{1-x^k}\color{blue}{=\prod_{k=1}^\infty\frac{1}{1-x^{2k-1}}} \end{align*}

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  • $\begingroup$ Thank you for your answer and comment, it's very clear! $\endgroup$ – luw Mar 4 '18 at 20:19
  • $\begingroup$ @luw: You're welcome. $\endgroup$ – Markus Scheuer Mar 4 '18 at 20:24

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