Binomial Expansion of 1 - x-n and 1/xn: A Comprehensive Guide

The binomial expansion is a fundamental concept in mathematics that allows us to express powers of binomials as a series. In this guide, we delve into the expansions of 1 - x-n and 1/xn. These expansions have wide-ranging applications in calculus, combinatorics, and other fields.

1 - x-n

The expansion of 1 - x-n can be expressed using the generalized binomial theorem:

1 - x-n u2211k0u221E u0393(nk-1, k) xk where u0393(nk-1, k) u221Fjnk-1nk-1-j / u221Fj0k-1

This series converges for x 1. The generalized binomial coefficient u0393(nk-1, k) can be defined as:

u0393(nk-1, k) u221Fjnk-1nk-1-j / u221Fj0k-1

1/xn

The expansion of 1/xn is given by the standard binomial theorem:

1/xn u2211k0n u0393(n, k) xn-k where u0393(n, k) n! / (k! (n-k)!)

This expansion is valid for any real number n and for x 1.

Summary

These expansions are incredibly useful in various applications including calculus and combinatorics. For instance, they can be used to approximate functions, solve differential equations, and simplify complex expressions.

Understanding the Coefficients

For the expansion of 1 - x-n, the coefficient of xr in the expansion is given by u0393(nr-1, r). Therefore, we have:

1 - x-n u2211r0u221E u0393(nr-1, r) xr

Similarly, for the expansion of 1/xn, the coefficient of xr is given by u0393(n, r). Therefore, we have:

1/xn u2211r0n u0393(n, r) xn-r

Note that 1 - x-n contains an infinite number of terms in its expansion, while 1/xn contains n 1 terms.

Conclusion

The binomial expansions of 1 - x-n and 1/xn offer powerful tools for mathematical analysis and problem-solving. By understanding and applying these expansions, mathematicians and students can simplify complex expressions and derive meaningful results in various fields.