Identify The Function Represented By The Following Power Series

Identifying the Function Represented by a Power Series

Power series are a fundamental tool in mathematical analysis, providing a way to represent functions as infinite sums of powers of a variable. Identifying the function represented by a given power series is a crucial skill, with applications spanning diverse fields like physics, engineering, and computer science. This article delves into the process of identifying these functions, exploring various techniques and providing illustrative examples.

Understanding Power Series

A power series is an infinite series of the form:

∑_n=0^∞ a_n(x - c)ⁿ = a₀ + a₁(x - c) + a₂(x - c)² + a₃(x - c)³ + ...

where:

• a_n are the coefficients of the series (constants).
• x is the variable.
• c is the center of the series (a constant).

The series converges for certain values of x and diverges for others. The set of x values for which the series converges is called the interval of convergence, and its radius is the radius of convergence.

Methods for Identifying the Function

Several methods exist to identify the function represented by a power series. These include:

1. Recognizing Known Series Expansions

The most straightforward approach involves recognizing the given power series as a known Taylor or Maclaurin series expansion of a common function. Many standard functions have well-established series representations. For example:

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

By comparing the given series with these known expansions, we can often directly identify the function. This requires familiarity with a range of common series.

2. Manipulating Known Series

Sometimes, the given series isn't directly recognizable but can be derived from a known series through algebraic manipulation. This might involve:

• Differentiation: Differentiating a known series term-by-term can yield a new series representing the derivative of the original function.
• Integration: Similarly, integrating a known series term-by-term can provide a series representing the integral of the original function.
• Substitution: Substituting a variable or expression into a known series can generate a new series.
• Multiplication or Division: Multiplying or dividing a known series by a simple expression can sometimes reveal the underlying function.

These manipulations require careful attention to the interval of convergence, as these operations can alter it.

3. Using the Formula for Taylor or Maclaurin Series

If the series isn't readily identifiable by the previous methods, we can attempt to determine the function by constructing its Taylor or Maclaurin series. The Taylor series of a function f(x) centered at c is given by:

f(x) = ∑_n=0^∞ f⁽ⁿ⁾(c)(x - c)ⁿ/n!

where f⁽ⁿ⁾(c) denotes the nth derivative of f(x) evaluated at x = c. If c = 0, this becomes the Maclaurin series. By calculating the derivatives of the function and evaluating them at the center, we can compare the resulting series with the given series to identify the function. This method is computationally intensive, especially for higher-order derivatives.

4. Using Partial Fraction Decomposition

For rational functions (ratios of polynomials), partial fraction decomposition can simplify the expression into a sum of simpler fractions. Each of these simpler fractions may then have a recognizable power series representation, making the identification of the function for the original series possible.

Illustrative Examples

Let's work through a few examples to solidify these concepts:

Example 1: Identify the function represented by the power series:

∑_n=0^∞ (xⁿ/2ⁿ)

This series resembles the geometric series ∑_n=0^∞ xⁿ = 1/(1-x) (|x| < 1). We can rewrite the given series as:

∑_n=0^∞ (x/2)ⁿ

This is a geometric series with a common ratio of x/2. Therefore, the function represented by this power series is:

f(x) = 1/(1 - x/2) = 2/(2 - x) (|x| < 2)

Example 2: Identify the function represented by the power series:

∑_n=1^∞ (-1)ⁿ⁺¹xⁿ/n

This series is the Maclaurin series for ln(1+x), valid for -1 < x ≤ 1. Therefore, the function is:

f(x) = ln(1+x)

Example 3: Identify the function represented by the power series:

∑_n=0^∞ (-1)ⁿx²ⁿ/(2n)!

This series is the Maclaurin series for cos(x). Therefore, the function is:

f(x) = cos(x)

Example 4 (requiring manipulation): Identify the function represented by the series:

∑_n=1^∞ nx^n-1

Recall the geometric series: ∑_n=0^∞ xⁿ = 1/(1-x) for |x| < 1. Differentiating both sides with respect to x, we get:

∑_n=1^∞ nx^n-1 = 1/(1-x)²

Therefore, the function represented by the given series is:

f(x) = 1/(1-x)² (|x| < 1)

Advanced Techniques and Considerations

For more complex power series, advanced techniques may be needed. These include:

• Ratio Test and Root Test: These tests determine the radius of convergence of the power series, providing information about the domain where the function is defined by the series.
• Abel's Theorem: This theorem addresses the convergence of the power series at the endpoints of the interval of convergence.
• Complex Analysis: Power series can be extended to the complex plane, providing powerful tools for analyzing functions with complex arguments and using techniques like contour integration.

Identifying the function represented by a power series often requires a blend of recognition, algebraic manipulation, and potentially, the construction of the Taylor or Maclaurin series. A strong foundation in calculus and series manipulation is essential for mastering this skill. With practice and familiarity with common series expansions, you'll become adept at unraveling the functions hidden within these infinite sums.