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Golden Ratio Series Sum

Let $ \tau = \frac{1 + \sqrt{5}}{2}, $ the golden ratio. Then \[\frac{1}{\tau} + \frac{1}{\tau^2} + \frac{1}{\tau^3} + \dotsb = \tau^n\]for some integer $ n $. Find $ n $.

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$\frac{a}{b}$
.
0
=
%

$a^n$
$a^{\circ}$
$a_n$
$\sqrt{}$
$\sqrt[n]{}$
$\pi$
$\ln{}$
$\log$
$\theta$
$\sin{}$
$\cos{}$
$\tan{}$

$($
$)$
$[$
$]$
$\cap$
$\cup$
$,$
$\infty$