Answer
$\dfrac{238}{24975}$
Work Step by Step
We are given the repeating decimal:
$0.00\overline{952}=0.00952952....$
Rewrite the given repeating decimal:
$0.00952952....=0.00952+0.0000952+0.000000952+.......$ $=\sum_{k=0}^{\infty} 0.00952\cdot 10^{-3k}$
The infinite geometric series $\sum_{k=0}^{\infty} a_1r^k$ has the sum $S=\dfrac{a_1}{1-r}$, where $r=0.001$.
Determine the value of the repeating decimal:
$\sum_{k=0}^{\infty} 0.00952\cdot 10^{-3k}=\dfrac{0.00952}{1-0.001}=\dfrac{\dfrac{952}{100000}}{\dfrac{999}{1000}}=\dfrac{952}{99900}=\dfrac{238}{24975}$
