To find the binomial expansion of \((1 - 4x)^{-1}\), we can use the binomial series formula for a negative exponent, which is given by:

\[
(1 + x)^n = \sum_{k=0}^{\infty} \binom{n}{k} x^k
\]

For our case, \( n = -1 \) and \( x = -4x \). Let's expand it:

\[
(1 - 4x)^{-1} = \sum_{k=0}^{\infty} \binom{-1}{k} (-4x)^k
\]

To compute this, we use the general term \(\binom{n}{k} = \frac{n(n-1)(n-2)\dots(n-k+1)}{k!}\). For \( n = -1 \), we get:

\[
\binom{-1}{k} = \frac{(-1)(-1-1)(-1-2)\dots(-1-(k-1))}{k!}
\]

\[
= \frac{(-1)(-2)(-3)\dots(-k)}{k!} = \frac{(-1)^k k!}{k!} = (-1)^k
\]

Thus, the expansion becomes:

\[
(1 - 4x)^{-1} = \sum_{k=0}^{\infty} (-1)^k (-4x)^k
\]

\[
= \sum_{k=0}^{\infty} 4^k x^k
\]

Hence, the binomial expansion of \((1 - 4x)^{-1}\) is:

\[
1 + 4x + 16x^2 + 64x^3 + 256x^4 + \dots
\]

This series converges for \(|4x| < 1\) or \(|x| < \frac{1}{4}\).

Would you like a detailed explanation of any specific part, or do you have any questions?

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### Relative Questions:

1. What is the general term in the binomial expansion of \((1 - 4x)^{-1}\)?
2. For which values of \(x\) does the series for \((1 - 4x)^{-1}\) converge?
3. How would you find the coefficient of \(x^3\) in the expansion of \((1 - 4x)^{-1}\)?
4. What is the binomial expansion of \((1 + 4x)^{-1}\)?
5. How does the expansion of \((1 + x)^{-2}\) differ from \((1 + x)^{-1}\)?
6. What is the radius of convergence of the series \(\sum_{k=0}^{\infty} 4^k x^k\)?
7. How would you derive the binomial series formula for a negative exponent?
8. Can the binomial series formula be applied to fractional exponents? How?

#### Tip:

When dealing with binomial expansions for negative or fractional exponents, ensure that the absolute value of the variable multiplied by the coefficient in the expression is less than 1 for convergence of the series.

Explore the binomial expansion of (1 - 4x)^-1 with a detailed step-by-step solution. Understand the binomial series formula for negative exponents and the conditions for convergence. Suitable for advanced high school students and anyone interested in series expansions.

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