# convergence of power series

We have given a power series $\sum\limits_{n=0}^\infty a_nx^n$ and $a_0\ne0$. Suppose the power series has radius of convergence $R>0$. Then we may assume that $a_0=1.$

Explanation: It's $$\sum\limits_{n=0}^\infty a_nx^n=a_0\sum\limits_{n=0}^\infty \frac {a_n}{a_0}x^n$$

But why do $\sum\limits_{n=0}^\infty \frac {a_n}{a_0}x^n$ and $\sum\limits_{n=0}^\infty a_nx^n$ have the same radius of convergence $R$ ?

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2022-07-25 20:46:47
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