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Question

# Find the Taylor series about the indicated center and determine the interval of convergence. $f(x)=\frac{1}{x}, c=1$

Solution

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The Taylor series of the function $f(x)$ about $x=x_{0}$ ( called the Maclaurin series if $x_{0}=0$ ) is given by:

$f(x)=\sum_{n=0}^{\infty} \frac{f^{(n)}\left(x_{0}\right)}{n !}\left(x-x_{0}\right)^{n}\tag{1}$

This series converges for all $x \in(x_{0}-r, x_{0}+r)$, where $r$ is the radius of convergence of the series.

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