Answer
convergent, $\quad $sum = $\displaystyle \frac{1}{9}$
Work Step by Step
An infinite geometric series is a series of the form
$ a+ar+ar^{2}+ar^{3}+\cdots+ar^{n-1}+\cdots$
An infinite geometric series for which $|r| < 1$
has the sum $S=\displaystyle \frac{a}{1-r}$
If $|r| \geq 1$, the series diverges (the sum does not exist).
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$a=0.1$
$r=\displaystyle \frac{0.01}{0.1}\times\frac{10}{10}=0.1, \quad |r| < 1,$
so the series is convergent,
$S=\displaystyle \frac{a}{1-r}=\frac{0.1}{1-0.1}=\frac{0.1}{0.9}=\frac{1}{9}$
