Answer
The series diverges.
Work Step by Step
The series can be rewritten as
$\mathop \sum \limits_{k = 1}^\infty {5^{3k}}\cdot{7^{1 - k}} = \mathop \sum \limits_{k = 1}^\infty {\left( {{5^3}} \right)^k}\cdot\frac{7}{{{7^k}}} = \mathop \sum \limits_{k = 1}^\infty 7{\left( {\frac{{{5^3}}}{7}} \right)^k}$
This is a geometric series with $r = \frac{{{5^3}}}{7}$. Since $\left| r \right| = \frac{{{5^3}}}{7} > 1$, the series diverges.
