Why is it that $\sum na_n x^{n-1}$ and $\sum na_n x^n$ have the same radius of convergence $R$?
How can I show that $$\lim \sup|na_n|^{1/n} = \lim \sup |na_n x|^{1/n}$$ since this would mean that both series have the same radius of convergence $R$?
2026-04-11 12:56:27.1775912187
Why do $\sum na_n x^{n-1}$ and $\sum na_n x^n$ have the same radius of convergence?
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Instead of using any formula for radius of convergence simply use the fact that $\sum na_nx^{n-1}$ converges iff $\sum na_nx^n$ does, for any $x \neq 0$. Obviously, this implies that they have the same radius of convergence.