To solve the given problem, let's go through the expansion of [tex]\((1 + 3x)^{-2}\)[/tex] and find the required information step-by-step:
1. Find the coefficient of [tex]\(x^2\)[/tex] in the expansion of [tex]\((1 + 3x)^{-2}\)[/tex]:
The coefficient of [tex]\(x^2\)[/tex] in the expansion of [tex]\((1 + 3x)^{-2}\)[/tex] is:
[tex]\( \boxed{0} \)[/tex]
2. Find the coefficient of [tex]\(x^3\)[/tex] in the expansion of [tex]\((1 + 3x)^{-2}\)[/tex]:
The coefficient of [tex]\(x^3\)[/tex] in the expansion of [tex]\((1 + 3x)^{-2}\)[/tex] is:
[tex]\( \boxed{0} \)[/tex]
3. Determine the values of [tex]\(x\)[/tex] for which the expansion is valid:
The expansion of [tex]\((1 + 3x)^{-2}\)[/tex] is a binomial series expansion, which converges within a certain range. In this case, the expansion is valid for:
[tex]\( -\frac{1}{3} < x < \frac{1}{3} \)[/tex]
Hence, the values of [tex]\(x\)[/tex] for which the series expansion is valid are:
[tex]\( \boxed{-\frac{1}{3}} < x < \boxed{\frac{1}{3}} \)[/tex]
So, the detailed step-by-step solution leads us to:
- The coefficient of [tex]\(x^2\)[/tex] in the expansion of [tex]\((1 + 3x)^{-2}\)[/tex] is [tex]\( 0 \)[/tex].
- The coefficient of [tex]\(x^3\)[/tex] in the expansion of [tex]\((1 + 3x)^{-2}\)[/tex] is [tex]\( 0 \)[/tex].
- The values of [tex]\(x\)[/tex] for which the expansion is valid are [tex]\(-\frac{1}{3} < x < \frac{1}{3}\)[/tex].
