# f(x)=2/(19+x). Find the power series representation and the interval of convergence.

$f\left(x\right)=\frac{2}{19+x}$
Find the power series representation and the interval of convergence.
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FieniChoonin

$f\left(x\right)=\frac{2}{19+x}$
$=\frac{\frac{2}{19}}{1+\frac{x}{19}}$
$=\frac{2}{19}\sum _{n=0}{\left(\frac{x}{19}\right)}^{n}$
$=\sum _{n=0}\left(2\frac{{x}^{n}}{{19}^{n+1}}\right)$
The power serires representation:
$f\left(x\right)=\sum _{n=0}\left(2\frac{{x}^{n}}{{19}^{n+1}}\right)$
he interval of convergence is:
$|\frac{x}{19}|<1\to -19
$x\in \left(-19,19\right)$