The ratio or root tests are the usual go-to methods.
By the ratio test, the series will converge if
\(\displaystyle\lim_{n\to\infty}\left|\dfrac{(n+1)!(4x-1)^{n+1}}{n!(4x-1)^n}\right|<1\)
The limit reduces to
\(|4x-1|\displaystyle\lim_{n\to\infty}|n+1|=\infty>1\)
which means the series diverges everywhere except the point \(x=\dfrac14\), where the series evaluates to 0. So because the series diverges, the interval of convergence is a single point, and so the radius of convergence is 0.
