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> A sequence $a\_1,a\_2,a\_3,\ldots$ is defined recursively by $a\_1 = 1$ and $a\_{2^k+j} = -a\_j \text{ } (j = 1,2,\ldots,2^k)$. Prove that this sequence is not periodic.
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We see that $$A(x) = \sum\_{n \ge 0} a\_n x^n = x(1 - x)(1 - x^2)(1 - x^4) \cdots.$$ Now for the sequence to be periodic, would $A(x)$ have to be representable as a rational function? How would we prove that? In that case we see that $A(x)$ can't be a rational function since it is $0$ at infinitely many points.