Understanding the Formula for 1 - x^-3: A Comprehensive Guide

Introduction

The mathematical expression 1 - x^{-3} can be expanded using various methods, including the binomial series expansion and the geometric series expansion. This guide explores both approaches and provides a deep understanding of how to derive and apply these formulas.

Binomial Series Expansion

General Formula

The general formula for the binomial expansion of 1 - x^{-n} is given by:

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

Application to 1 - x^-3

When n 3, this becomes:

1 - x^{-3} sum_{k0}^{infty} binom{3}{k} x^k 1 3x 6x^2 1^3 ldots

The binomial coefficient binom{3}{k} can be calculated as:

binom{3}{k} frac{3 2k 1}{2k}

Series Expansion

The series expansion is as follows:

1 - x^{-3} 1 3x 6x^2 1^3 ldots

This series converges for x .

Geometric Series Expansion

Geometric Series for 1 / (1 - x^3)

If we consider the expression 1 / (1 - x^3), it can be expanded as a geometric series:

1 / (1 - x^3) 1 x^3 x^6 x^9 ldots

By substituting x^3 into the binomial expansion, we obtain:

1 - x^{-3} 1 3x 6x^2 1^3 ldots

Derivation Using Differentiation

Deriving the Series Using the Binomial Theorem

The formula can also be derived using the binomial theorem. By considering the series expansion of:

1 - x^{-3} 1 - x x^2 - x^3 x^4 - ldots

and then expanding each term, we get:

1 - x^{-3} 1 3x 6x^2 1^3 ldots

Conclusion

Understanding and applying the formulas for the expression 1 - x^{-3} is crucial in various mathematical and engineering applications. Whether through the binomial series expansion or the geometric series approach, these methods provide a robust foundation for further exploration and analysis.