Binomial Expansion for |x| < 1
The binomial theorem provides a powerful tool to expand expressions of the form $(a+x)^n$ where n is a positive integer. When dealing with expressions where |x| < 1, we can utilize the binomial theorem to create an infinite series representation.
The Binomial Theorem
For any real number x such that |x| < 1 and any real number n, we have:
(1 + x)^n = 1 + nx + (n(n-1)/2!)x² + (n(n-1)(n-2)/3!)x³ + ...
This is an infinite series where each term is obtained by the following formula:
(n(n-1)(n-2)...(n-k+1)/k!)x^k
where k is a non-negative integer.
Understanding the Formula
- n(n-1)(n-2)...(n-k+1): This represents the product of k consecutive numbers starting from n and decreasing by 1.
- k!: This represents the factorial of k, which is the product of all positive integers less than or equal to k.
- x^k: This is the power of x in the current term.
Convergence of the Series
The series representation of (1+x)^n converges for |x| < 1. This means that as we add more terms to the series, the sum gets closer and closer to the actual value of (1+x)^n.
Applications
The binomial expansion for |x| < 1 has numerous applications in various fields, including:
- Calculus: To approximate functions, differentiate and integrate functions, and solve differential equations.
- Probability and Statistics: To calculate probabilities in binomial distributions and analyze statistical data.
- Physics and Engineering: To model and solve problems related to motion, waves, and electricity.
Example
Let's consider an example:
(1 + x)^(-1)
Using the binomial theorem, we can expand this as:
(1 + x)^(-1) = 1 + (-1)x + (-1)(-2)/2! x² + (-1)(-2)(-3)/3! x³ + ...
This simplifies to:
(1 + x)^(-1) = 1 - x + x² - x³ + ...
This infinite series converges for |x| < 1 and represents the function 1/(1+x).
Conclusion
The binomial expansion for |x| < 1 is a powerful tool that allows us to express functions in a series form, leading to numerous applications in various fields. Understanding its derivation and convergence properties is crucial for effectively utilizing this mathematical concept.
