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Given a power series of the form

$$f(z)=\sum_{n=0}^\infty a_nz^n$$ with $z\in\mathbb{C}$ and radius of convergence $R$, then itss derivatives is $$f’(z)=\sum_{n=1}^\infty n\,a_nz^{n-1}$$ with radius of convergence $R$.

Now, some texts define the derivative as $$f’(z)=\sum_{n=0}^\infty n\,a_nz^{n-1}$$ So my questions is, why do some texts use one and not the other? Which is the preferred use?

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  • $\begingroup$ The expression $z^{n-1}$ is not defined for $z=0$ and $n=0$. $\endgroup$ – egreg Feb 24 '18 at 23:57
  • $\begingroup$ @egreg so the second form would be wrong it stated just like that? Because many well-known books use it $\endgroup$ – user372003 Feb 25 '18 at 0:46
  • $\begingroup$ Some use the convention that “integer $0$ times undefined gives $0$”, but pay attention that removing the zero term is not always right. Think to the power series for the sine. $\endgroup$ – egreg Feb 25 '18 at 6:01
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Basicaly it is same thing.If we start in last series with $n=0$ we would get first term $0$.That means we can start from $n=1$

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