Answer
$10$
Work Step by Step
We will consider the theorem of convergence of an Infinite Geometric Series in order to solve this probelm.
The infinite geometric series $\displaystyle \sum_{k=1}^{\infty}a_{1}r^{k-1}$ converges when $|r| \lt 1,$and then its sum can be calculated as:
$\displaystyle \sum_{k=1}^{\infty}a_{1}r^{k-1}=\frac{a_{1}}{1-r}=S_\infty$
where $r$ is the common ratio.
Since $r=0.90 \lt 1$, this shows that the infinite geometric series converges.
Its sum will be :
$S_\infty=\dfrac{a_{1}}{1-r}=\displaystyle \frac{1}{1-0.90}=10$
