%5E(-2)-expansion.jpeg "(1-x)^(-2) Expansion")
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:
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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.