taylor expansion of 1/1 x

Natashya

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17-03-2025

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The Taylor expansion, or Taylor series, is a powerful tool in calculus for approximating the value of a function at a specific point using its derivatives at another point. This article will delve into the Taylor expansion of the function 1/(1-x), demonstrating its derivation and showcasing its applications. Understanding this expansion is fundamental to many areas of mathematics, physics, and engineering.

Understanding the Taylor Expansion

The Taylor expansion of a function f(x) around a point a is given by:

f(x) = f(a) + f'(a)(x-a)/1! + f''(a)(x-a)²/2! + f'''(a)(x-a)³/3! + ...

This infinite sum represents the function f(x) as a series of terms involving its derivatives at point a and powers of (x-a). The factorial terms (1!, 2!, 3!, etc.) are there for normalization. The accuracy of the approximation improves as more terms are included.

Deriving the Taylor Expansion of 1/(1-x)

Let's find the Taylor expansion of f(x) = 1/(1-x) around the point a = 0. This is also known as the Maclaurin series.

First, we need the derivatives of f(x):

• f(x) = (1-x)⁻¹
• f'(x) = (1-x)⁻²
• f''(x) = 2(1-x)⁻³
• f'''(x) = 6(1-x)⁻⁴
• and so on...

Evaluating these derivatives at a = 0:

• f(0) = 1
• f'(0) = 1
• f''(0) = 2
• f'''(0) = 6

Notice a pattern emerging: the nth derivative evaluated at 0 is n!

Substituting these into the Taylor expansion formula:

f(x) = 1 + x + 2x²/2! + 6x³/3! + ... = 1 + x + x² + x³ + ...

This simplifies to the well-known geometric series:

1/(1-x) = Σ (xⁿ)  for |x| < 1.

The condition |x| < 1 is crucial. The geometric series only converges (meaning the sum approaches a finite value) when the absolute value of x is less than 1. Otherwise, the series diverges.

Applications of the Taylor Expansion of 1/(1-x)

The Taylor expansion of 1/(1-x) has numerous applications:

• Calculating Approximations: For values of x close to 0, the first few terms of the series provide a good approximation of 1/(1-x). This is particularly useful when dealing with computationally expensive calculations.

• Solving Differential Equations: This expansion often appears as a solution or part of the solution to various differential equations, particularly those encountered in physics and engineering.

• Generating Other Taylor Expansions: By manipulating the series (e.g., integration or differentiation), we can derive Taylor expansions for other functions. For example, integrating the series term-by-term gives the Taylor expansion for -ln(1-x).

• Probability and Statistics: The geometric series, representing the expansion of 1/(1-x), underlies many probability distributions and calculations.

Limitations and Considerations

Remember, the Taylor expansion is only valid within its radius of convergence (|x| < 1 in this case). Outside this interval, the series does not accurately represent the function. Using it beyond this range would lead to incorrect results. Also, the further x is from 0, the more terms are needed for a decent approximation.

Conclusion

The Taylor expansion of 1/(1-x) is a fundamental result in calculus with wide-ranging applications. Its derivation, based on the general Taylor expansion formula, showcases the elegance and power of this mathematical tool. Understanding this expansion provides a solid foundation for tackling more advanced topics in calculus and its applications. Always remember to check the radius of convergence to ensure the validity of your approximations.