Maclaurin Series and Power Series - Convergence and Validity of Power Series

Find the range of values of \(x\) for which the expansion is valid for the following:

1. \({{(1-3x)}^{-2}}\)
2. \({{(1-\frac{x}{4})}^{-3}}\)
3. \({{(2+5x)}^{-3}}\)
4. \({{(3+\frac{x}{7})}^{\frac{1}{2}}}\)
5. \({{(4+9{{x}^{2}})}^{-2}}\)
6. \({{(1+\frac{3}{{{x}^{2}}})}^{-5}}\)
7. \(\frac{\sqrt{2-x}}{{{(x-1)}^{2}}}\)
8. \(\frac{x-6}{{{(x-6)}^{2}}\sqrt{3x-2}}\)