(1-x)^(-2) Expansion

3 min read Jun 16, 2024

The Binomial Expansion of (1-x)^(-2)

The expression (1-x)^(-2) is a binomial raised to a negative power. We can use the binomial theorem to expand this expression. The binomial theorem states:

(1 + x)^n = 1 + nx + n(n-1)x^2/2! + n(n-1)(n-2)x^3/3! + ...

where 'n' can be any real number (including negative numbers).

Expanding (1-x)^(-2)

To apply the binomial theorem to (1-x)^(-2), we need to replace 'x' with '-x' and set 'n' to -2.

Let's break down the expansion step-by-step:

Substitute:
(1 - x)^(-2) = 1 + (-2)(-x) + (-2)(-3)(-x)^2/2! + (-2)(-3)(-4)(-x)^3/3! + ...
Simplify: (1 - x)^(-2) = 1 + 2x + 3x^2 + 4x^3 + ...

Recognizing the Pattern

Notice that the coefficients of the expansion follow a pattern:

The constant term is 1.
The coefficient of x is 2.
The coefficient of x^2 is 3.
The coefficient of x^3 is 4, and so on.

Therefore, the general term in the expansion can be expressed as:

(n+1)x^n

Writing the Complete Expansion

The complete expansion of (1-x)^(-2) can be written in summation form:

(1 - x)^(-2) = Σ (n+1)x^n , where n = 0, 1, 2, 3, ...

Important Notes

The expansion of (1-x)^(-2) is an infinite series. This means it has an infinite number of terms.
The expansion is valid only for |x| < 1. For values of x outside this range, the series diverges and doesn't converge to a finite value.

Applications

The binomial expansion of (1-x)^(-2) has applications in various fields such as:

Calculus: It can be used to find derivatives of functions like (1-x)^(-2).
Statistics: It's used in calculating probabilities in certain statistical distributions.
Physics: It appears in solving certain physical problems involving forces and potentials.

Understanding the expansion of (1-x)^(-2) provides a powerful tool for working with this expression in various mathematical and scientific contexts.