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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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Just go back to the definition of the radius of convergence of a power series: it's a number $R$ such that the series $\sum_{n=0}^{+\infty}a_nx^n$ is convergent if $|x|. Since the convergence of $\sum_{n=0}^{+\infty}a_nx^n$ and $\frac 1{a_0}\sum_{n=0}^{+\infty}a_nx^n$ are equivalent, it gives the result.

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    Well, if you are given only the radius of convergence, then you may assign any arbitrary values to the first finitely many coefficients. Because R= 1/(lim sup |a(n)|^(1/n)) and lim sup of a sequence does not depend on the first finitely many terms of it!2012-05-13