Notice that the left hand side is a G.P. with first term = 1/r and common ration = r.
The left hand side is an infinite sum which only converges when the variable is in a certain bound.
The bound is -1 < r < 1. Otherwise it will grow to infinity very quickly.
Having that in mind, let's use the formula with the contraints -1 < r < 1.
Sum of infinite G.P. = \(\dfrac{a}{1 - r}\), where a is the first term and r is the common ratio.
So,
\(\dfrac{\dfrac1r}{1 - r} = \dfrac4r\\ \dfrac{\dfrac1{r} \cdot\color{blue}r\color{black}}{1 - r} = \dfrac{4\color{blue}r\color{black}}r\\ \dfrac1{1 - r}= 4\\ 1 - r = \dfrac14 \)
I believe you can continue from here.