Find rational notation for $4.\overline{125}$ using the concept of a series

Question

Find rational notation for $4.\overline{125}$ using the concept of a series.

I know how to find rational notation using the cheap trick called putting the repeating digits over $9$'s, getting $\dfrac{4121}{999}$, but how can I do it using the concept of an infinite geometric series?

Answer 1

Try this:

$$4.\overline{125} = 4 + 0.125 + 0.000125 + 0.000000125 + \ldots\\ = 4 + \frac{125}{1000} + \frac{125}{1000^2} + \frac{125}{1000^3} + \ldots$$

And now you can factor out $\frac{125}{1000}$ and apply what you know about $\sum_{k=0}^{\infty}r^k$.