Understanding and Calculating the Binomial Expansion of 1/x^-n using the Generalized Binomial Theorem

The binomial expansion for 1/x^-n, or more generally (1 x)^{-n}, can be derived using the generalized binomial theorem. This theorem is particularly useful for expanding expressions where the exponent is negative, and it provides a powerful tool in algebra and calculus.

What is the Generalized Binomial Theorem?

The generalized binomial theorem provides the expansion of (1 x)^n for any real number n, which can be written as:

(1 x)^n sum_{k0}^{infty} binom{n}{k} x^k

Applying the Generalized Binomial Theorem to 1/x^-n

For the specific case of 1/x^-n, we need to adjust the form slightly. Let's consider the expression (1 (-x))^n, which is equivalent to 1/x^-n. This expansion can be derived as:

1/x^-n (1 (-x))^-n sum_{k0}^{infty} binom{-n}{k} (-x)^k

The generalized binomial coefficient binom{-n}{k} is defined as:

binom{-n}{k} frac{-n(-n-1)(-n-2)cdots(-n-k 1)}{k!}

Steps to Calculate the Expansion

Identify n: Determine the value of n for your specific case. Write the General Term: The general term in the expansion is:

T_k binom{-n}{k} (-x)^k frac{-n(-n-1)(-n-2)cdots(-n-k 1)}{k!} (-x)^k

Sum the Series: The full expansion can be written as:

1/x^-n sum_{k0}^{infty} frac{-n(-n-1)(-n-2)cdots(-n-k 1)}{k!} (-x)^k

Example

Let's consider an example where n 2. The expansion becomes:

1/x^-2 (1 (-x))^-2 sum_{k0}^{infty} binom{-2}{k} (-x)^k

For k 0: T_0 binom{-2}{0} (-x)^0 1 For k 1: T_1 binom{-2}{1} (-x)^1 -2(-x) 2x For k 2: T_2 binom{-2}{2} (-x)^2 frac{(-2)(-3)}{2!} x^2 3x^2

Thus, the first few terms of the expansion are:

1/x^-2 1 2x 3x^2 - 4x^3 ...

Convergence

This series converges for |x|

Additional Formulations

There are also simpler formul? that you can use for the binomial expansion of 1/x^-n:

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

This can be written more succinctly as:

1/x^-n 1 - nx frac{n(n-1)}{2!}x^2 - frac{n(n-1)(n-2)}{3!}x^3 ...

Alternatively, for the specific case n 4, the expansion is:

1/x^-4 1 - 4x 6x^2 - 4x^3 x^4

Conclusion

I hope this helps in understanding how to calculate the binomial expansion of 1/x^-n. If you have any questions, feel free to ask!