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?

---

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.