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Power Serieshard
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Let f(x)=∑n=0∞xnf(x) = \sum_{n=0}^{\infty} x^nf(x)=∑n=0∞​xn for ∣x∣<1|x| < 1∣x∣<1 (so f(x)=11−xf(x) = \frac{1}{1-x}f(x)=1−x1​). If ∣g(x)∣<1|g(x)| < 1∣g(x)∣<1 for all ∣x∣<r|x| < r∣x∣<r, what is the radius of convergence of the composed series f(g(x))=∑n=0∞g(x)nf(g(x)) = \sum_{n=0}^{\infty} g(x)^nf(g(x))=∑n=0∞​g(x)n?