taylor expansion of 1/1 x

Daniel Parker

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16-03-2025

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The function 1/(1-x) is a cornerstone in calculus and has a particularly elegant Taylor expansion, which is incredibly useful in various applications, from approximating complex functions to deriving other series. This article will explore the derivation and applications of this expansion.

Understanding Taylor Expansions

Before diving into the specific expansion of 1/(1-x), let's briefly recap the concept of Taylor expansions. A Taylor expansion represents a function as an infinite sum of terms, each involving a derivative of the function at a specific point (typically zero) and a power of (x-a), where 'a' is the point of expansion. For the point of expansion a=0, this is also known as a Maclaurin series. The general formula is:

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

The accuracy of the approximation improves as more terms are included in the sum. The radius of convergence defines the range of x values for which the series converges to the function's actual value.

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

Let's find the Taylor expansion for f(x) = 1/(1-x) around a = 0 (Maclaurin series). We need to calculate the derivatives of f(x):

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

Evaluating these at x = 0:

• f(0) = 1
• f'(0) = 1
• f''(0) = 2
• f'''(0) = 6
• ... Notice a pattern: the nth derivative at x=0 is n!

Substituting these into the Taylor expansion formula:

1/(1-x) = 1 + x + 2x²/2! + 6x³/3! + 24x⁴/4! + ...

Simplifying:

1/(1-x) = 1 + x + x² + x³ + x⁴ + ...

This is a geometric series with the first term a = 1 and common ratio r = x. The formula for the sum of an infinite geometric series is a/(1-r), which converges to 1/(1-x) when |x| < 1 (the radius of convergence).

Applications of the Taylor Expansion

The Taylor expansion of 1/(1-x) has a wide range of applications:

• Approximating Functions: For values of x close to 0 and within the radius of convergence, the series provides an accurate approximation of 1/(1-x). Truncating the series after a certain number of terms gives a polynomial approximation.

• Solving Differential Equations: This expansion is often used in the solution of differential equations, especially those involving power series methods.

• Deriving Other Series: The expansion can be used to derive the Taylor expansions of other functions through substitution and manipulation. For example, by substituting -x² for x, we get the series expansion for 1/(1+x²).

• Probability and Statistics: This series is fundamental in probability theory and statistics, appearing in calculations involving probability distributions and generating functions.

• Physics and Engineering: This expansion finds its way into numerous physics and engineering applications, from analyzing electrical circuits to modelling physical phenomena.

Limitations and Considerations

It's crucial to remember that the Taylor expansion of 1/(1-x) only converges for |x| < 1. Outside this interval, the series diverges and does not provide a valid representation of the function. This constraint highlights the importance of understanding the radius of convergence when using Taylor series approximations.

Conclusion

The Taylor expansion of 1/(1-x) is a powerful tool with widespread applications across numerous fields. Its simplicity and elegant form make it a cornerstone in various mathematical and scientific computations. Understanding its derivation and limitations is essential for effectively using this powerful series in problem-solving. Remember to always check the radius of convergence to ensure the validity of your approximation.