I'm guessing the series is supposed to be
[tex]\displaystyle\sum_{n=1}^\infty\frac{n^2x^n}{7\cdot14\cdot21\cdot\cdots\cdot(7n)}[/tex]
By the ratio test, the series converges if the following limit is less than 1.
[tex]\displaystyle\lim_{n\to\infty}\left|\frac{\frac{(n+1)^2x^{n+1}}{7\cdot14\cdot21\cdot\cdots\cdot(7n)\cdot(7(n+1))}}{\frac{n^2x^n}{7\cdot14\cdot21\cdot\cdots\cdot(7n)}}\right|[/tex]
The first [tex]n[/tex] terms in the numerator's denominator cancel with the denominator's denominator:
[tex]\displaystyle\lim_{n\to\infty}\left|\frac{\frac{(n+1)^2x^{n+1}}{7(n+1)}}{n^2x^n}\right|[/tex]
[tex]|x^n|[/tex] also cancels out and the remaining factor of [tex]|x|[/tex] can be pulled out of the limit (as it doesn't depend on [tex]n[/tex]).
[tex]\displaystyle|x|\lim_{n\to\infty}\left|\frac{\frac{(n+1)^2}{7(n+1)}}{n^2}\right|=|x|\lim_{n\to\infty}\frac{|n+1|}{7n^2}=0[/tex]
which means the series converges everywhere (independently of [tex]x[/tex]), and so the radius of convergence is infinite.
