= Sum of power series with unequal radii of convergence
{title2=$R\ne S\ \Longrightarrow\ R_{\rm sum}=\min(R,S)$}
Inside both convergence discs, the <triangle inequality> gives <absolute convergence> of the sum. Suppose $R<S$ and the sum had radius $T>R$. At any modulus between $R$ and $\min(T,S)$, subtracting the absolutely convergent second series from the absolutely convergent sum would give <absolute convergence> of the first series outside its <radius of convergence>, a contradiction. Interchanging the two series covers $S<R$.
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