Find The Function Represented By The Power Series

Finding the Function Represented by a Power Series

Finding the function represented by a power series is a fundamental concept in calculus and analysis. It bridges the gap between discrete sequences and continuous functions, allowing us to represent complex functions using simpler, infinite series. This process involves recognizing the pattern in the series and relating it to known Taylor or Maclaurin series expansions. This article will explore various techniques for tackling this problem.

Understanding Power Series

A power series is an infinite series of the form:

∑_n=0^∞ c_n(x - a)ⁿ = c₀ + c₁(x - a) + c₂(x - a)² + ...

where:

• c_n are the coefficients of the series.
• x is the variable.
• a is the center of the series (often 0, leading to a Maclaurin series).

The series converges for certain values of x within its radius of convergence. Outside this radius, the series diverges. Determining the radius of convergence is a crucial first step in analyzing a power series.

Methods for Finding the Represented Function

Several methods can be used to identify the function represented by a given power series:

1. Recognizing Known Taylor/Maclaurin Series

This is the most straightforward approach. Familiarize yourself with the Taylor/Maclaurin series expansions of common functions like:

• e^x: ∑_n=0^∞ xⁿ/n!
• sin(x): ∑_n=0^∞ (-1)ⁿx²ⁿ⁺¹/(2n+1)!
• cos(x): ∑_n=0^∞ (-1)ⁿx²ⁿ/(2n)!
• 1/(1-x): ∑_n=0^∞ xⁿ (geometric series, |x| < 1)
• ln(1+x): ∑_n=1^∞ (-1)ⁿ⁺¹xⁿ/n (|x| ≤ 1, x ≠ -1)

By comparing the given power series to these known expansions, you can often identify the function directly, potentially with some manipulation such as substitution or differentiation/integration.

2. Manipulating Known Series

Sometimes, a power series might not directly match a known expansion, but it can be obtained through manipulations like:

• Substitution: Replace x with a function of x. For example, if you have ∑_n=0^∞ (x²)ⁿ/n!, you can substitute u = x² to recognize it as the series for e^u = e^x².
• Differentiation/Integration: Term-by-term differentiation or integration of a known series can yield a new series representing the derivative or integral of the original function. This is valid within the radius of convergence.
• Algebraic Manipulation: Simple algebraic operations like factoring or expanding terms can sometimes reveal a recognizable pattern.

3. Using the Formula for Taylor/Maclaurin Coefficients**

If the above methods fail, you can use the definition of Taylor or Maclaurin series coefficients:

c_n = f⁽ⁿ⁾(a)/n!

where f⁽ⁿ⁾(a) is the nth derivative of the function evaluated at a. By calculating several derivatives and comparing them to the given coefficients, you might deduce a pattern and identify the function. This method is often more computationally intensive.

Example:

Let's consider the power series:

∑_n=0^∞ x²ⁿ/n!

This resembles the Maclaurin series for e^x, but with x² instead of x. Therefore, we can substitute u = x², resulting in:

∑_n=0^∞ uⁿ/n! = e^u = e^x²

Thus, the power series represents the function e^x².

Conclusion:

Finding the function represented by a power series requires a systematic approach combining pattern recognition, knowledge of standard series expansions, and skillful manipulation of series. Mastering these techniques is essential for a strong understanding of calculus and its applications. Remember to always consider the radius of convergence, ensuring the function representation is valid within the appropriate interval.