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How do i find the power series of the form: $$\sum_{n=0}^{\infty}a_n ({z-z_0)}$$

where $f(z)=e^z$ and $z_0=1$

using the geomatric series currently i have that it equals

$$\sum_{n=0}^{\infty} ({e^z-1)}$$

However i dont believe this is correct since it in terms of $e^z$ rather than $z$

any help would be appreciated.

sean
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  • There is a standard formula for a power series centered at a point $a$: $f(z) = \sum_{n=0}^{\infty}f^{(n)}(a) (z-a)^{n}/n!$. Is there any particular reason why this has been tagged with complex analysis? – JessicaK Apr 02 '15 at 12:42
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    By $f(x)$, do you mean $z$? Because $f$ isn't in your summand. And neither is $x$, for that matter. – GFauxPas Apr 02 '15 at 12:55

1 Answers1

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you can just use : $$ e^z=e\cdot e^{z-1}$$ and then compute the series expansion on $0$ and you obtain: $$e^z=\sum_{n=0}^{+\infty}e\cdot\frac{(z-1)^n}{n!} $$

jameselmore
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Elaqqad
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