Answer
$$\dfrac{532}{99}$$
Work Step by Step
We have: $5.373737......=5+0.37+0.0037+....=5+\Sigma_{n=1}^{\infty} 0.37 (0.01)^{n-1}$
This shows a geometric series with common ratio $r=0.01$ and with initial term $a=0.37$. We see that $|r| \lt 1$, so the series converges.
Therefore, the sum is: $$5.373737=5+\dfrac{a}{1-r}\\=\dfrac{0.37}{1-0.01}\\=\dfrac{532}{99}$$
