Identify The Function Represented By The Following Power Series

Let's unravel the mystery of power series and how to identify the functions they represent. Power series are a cornerstone of mathematical analysis, providing a way to represent functions as infinite sums of terms involving powers of a variable. Understanding how to work with them is crucial for various applications in calculus, differential equations, and complex analysis.

Understanding Power Series: A Foundation

At its core, 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 + c₃(x - a)³ + ...

where:

• x is a variable.
• a is a constant, known as the center of the power series.
• c_n are constants, called the coefficients of the power series.

The key idea is that this infinite sum can define a function f(x) within a certain interval of x values. That interval is called the interval of convergence. The primary goal is to determine what function f(x) the power series represents.

Why are Power Series Important?

Power series offer several advantages:

• Representation of Functions: They allow us to represent a wide range of functions, including those that might not have simple closed-form expressions.
• Approximation: They can be used to approximate function values, especially when evaluating functions is computationally difficult. Truncating the series after a certain number of terms provides a polynomial approximation.
• Differentiation and Integration: Power series can be differentiated and integrated term-by-term within their interval of convergence, making them powerful tools for solving differential equations.
• Solving Differential Equations: Many differential equations can be solved by expressing the solution as a power series and determining the coefficients.
• Analyzing Complex Functions: Power series extend to complex analysis, where they are used to define analytic functions and explore their properties.

Identifying the Function Represented by a Power Series: A Step-by-Step Guide

The process of identifying the function represented by a power series involves a combination of pattern recognition, manipulation, and comparison with known series. Here's a structured approach:

1. Recognize Common Power Series:

Familiarity with the power series representations of common functions is essential. Memorizing these series and their intervals of convergence will significantly speed up the identification process. Here are some key examples:

• Geometric Series:

  1 / (1 - x) = ∑_n=0^∞ xⁿ for |x| < 1

  This is arguably the most important power series. It serves as the foundation for many other series.

• Exponential Function:

  e^x = ∑_n=0^∞ xⁿ / n! for all x

• Sine Function:

  sin(x) = ∑_n=0^∞ (-1)ⁿ x²ⁿ⁺¹ / (2n+1)! for all x

• Cosine Function:

  cos(x) = ∑_n=0^∞ (-1)ⁿ x²ⁿ / (2n)! for all x

• Natural Logarithm:

  ln(1 + x) = ∑_n=1^∞ (-1)ⁿ⁺¹ xⁿ / n for -1 < x ≤ 1

• Arctangent Function:

  arctan(x) = ∑_n=0^∞ (-1)ⁿ x²ⁿ⁺¹ / (2n+1) for |x| ≤ 1

2. Manipulation Techniques:

Often, the given power series won't perfectly match one of the standard forms. You'll need to manipulate the series to make it recognizable. Common manipulation techniques include:

• Substitution: Replace x with a function of x, such as x², -x, or 2x. For example, to find the power series for e^-x², substitute -x² for x in the power series for e^x.
• Multiplication by a Constant or Power of x: Multiply the entire series by a constant or a power of x. This will change the coefficients and potentially the starting index of the summation.
• Differentiation: Differentiate the power series term-by-term. Remember to adjust the index of summation accordingly. This can often lead to a recognizable form.
• Integration: Integrate the power series term-by-term. Don't forget the constant of integration! This can also help reveal the function.
• Separating the Series: Sometimes, separating the series into two or more series can make it easier to identify. For example, you might separate even and odd terms.
• Index Shifting: Adjusting the index of summation can be crucial for aligning terms and recognizing patterns. For example, if you have a series ∑_n=1^∞, you might want to rewrite it as ∑_n=0^∞ by replacing n with n+1 in the expression inside the summation.

3. Example Walkthroughs:

Let's illustrate the process with some examples:

Example 1: Identify the function represented by the power series ∑_n=0^∞ xⁿ / 3ⁿ.

• Recognize the pattern: This looks like a geometric series. We can rewrite it as ∑_n=0^∞ (x/3)ⁿ.
• Apply the geometric series formula: Since ∑_n=0^∞ rⁿ = 1 / (1 - r) for |r| < 1, we can identify r = x/3.
• Substitute: Therefore, the power series represents the function f(x) = 1 / (1 - x/3) = 3 / (3 - x).
• Determine the interval of convergence: The geometric series converges when |x/3| < 1, which means |x| < 3. The interval of convergence is -3 < x < 3.

Example 2: Identify the function represented by the power series ∑_n=1^∞ n x^n-1.

• Recognize the pattern: This doesn't directly match any of our standard series. However, the n in front of the x^n-1 suggests differentiation.

• Relate to a known series: Consider the geometric series ∑_n=0^∞ xⁿ = 1 / (1 - x) for |x| < 1.

• Differentiate: Differentiating both sides with respect to x gives:

  d/dx [∑_n=0^∞ xⁿ] = d/dx [1 / (1 - x)]

  ∑_n=1^∞ n x^n-1 = 1 / (1 - x)²

• Identify the function: Therefore, the power series represents the function f(x) = 1 / (1 - x)².

• Interval of convergence: The interval of convergence remains |x| < 1, or -1 < x < 1, since differentiation does not change the radius of convergence.

Example 3: Identify the function represented by the power series ∑_n=0^∞ (-1)ⁿ x²ⁿ / n!.

• Recognize the pattern: The (-1)ⁿ and the n! suggest a connection to the exponential function. The x²ⁿ indicates a substitution.

• Relate to a known series: Recall that e^x = ∑_n=0^∞ xⁿ / n!.

• Substitute: Substitute -x² for x in the exponential series:

  e^-x² = ∑_n=0^∞ (-x²)ⁿ / n! = ∑_n=0^∞ (-1)ⁿ x²ⁿ / n!

• Identify the function: Therefore, the power series represents the function f(x) = e^-x².

• Interval of convergence: The exponential function converges for all x, so the interval of convergence is (-∞, ∞).

Example 4: Identify the function represented by the power series ∑_n=0^∞ x²ⁿ⁺¹ / (2n+1).

• Recognize the pattern: The form x²ⁿ⁺¹ / (2n+1) resembles the arctangent series, but without the alternating sign. This suggests integration might be useful.

• Consider Differentiation: Let's differentiate the series term by term:

  d/dx [∑_n=0^∞ x²ⁿ⁺¹ / (2n+1)] = ∑_n=0^∞ x²ⁿ

• Relate to a known series: We can recognize this as a geometric series: ∑_n=0^∞ (x²)ⁿ = 1 / (1 - x²) for |x²| < 1, or |x| < 1.

• Integrate: Now, integrate the result with respect to x:

  ∫ [1 / (1 - x²)] dx = ∫ ∑_n=0^∞ x²ⁿ dx = ∑_n=0^∞ x²ⁿ⁺¹ / (2n+1) + C

  The integral of 1/(1-x²) can be solved using partial fractions: 1/(1-x²) = (1/2) [1/(1+x) + 1/(1-x)]. Therefore, ∫ [1 / (1 - x²)] dx = (1/2) [ln(1+x) - ln(1-x)] + C = (1/2) ln[(1+x)/(1-x)] + C

• Determine the constant of integration: When x = 0, the original series is 0. So, (1/2) ln[(1+0)/(1-0)] + C = 0, which means C = 0.

• Identify the function: Therefore, the power series represents the function f(x) = (1/2) ln[(1+x)/(1-x)].

• Interval of convergence: The interval of convergence is |x| < 1, or -1 < x < 1. We also need to check the endpoints. The series converges for x=0, but the convergence at x=+-1 needs to be checked separately, but the initial geometric series condition restricts the interval.

4. The Importance of the Interval of Convergence:

It is crucial to determine the interval of convergence for the power series. The power series only represents the function within this interval. The radius of convergence, R, is a non-negative real number or ∞ such that the series converges if |x - a| < R and diverges if |x - a| > R. The interval of convergence is (a - R, a + R), (a - R, a + R], [a - R, a + R), or [a - R, a + R], depending on the convergence at the endpoints.

Methods for determining the radius of convergence include:

• Ratio Test: Let a_n = c_n(x - a)ⁿ. Then, R = lim_n→∞ |a_n / a_n+1|, if the limit exists.
• Root Test: R = 1 / lim_n→∞ |c_n|^1/n, if the limit exists.

5. Common Mistakes to Avoid:

• Forgetting the Constant of Integration: When integrating a power series, always remember to add the constant of integration and determine its value using initial conditions.
• Ignoring the Interval of Convergence: The power series representation is only valid within its interval of convergence. Make sure to state the interval.
• Incorrectly Manipulating the Series: Be careful when performing algebraic manipulations. Double-check your work to avoid errors.
• Not Recognizing Basic Series: Spend time memorizing the power series representations of common functions. This will make the identification process much easier.
• Assuming Term-by-Term Differentiability/Integrability Always Holds: Power series can be differentiated and integrated term-by-term within their interval of convergence. You can't blindly apply these operations without considering convergence.

Advanced Techniques and Special Cases

While the above steps cover many common power series, some cases require more advanced techniques:

• Taylor and Maclaurin Series: If you suspect that a function f(x) can be represented by a power series, you can find its Taylor series (centered at a) or Maclaurin series (centered at 0). The Taylor series is given by:

  f(x) = ∑_n=0^∞ f⁽ⁿ⁾(a) (x - a)ⁿ / n!

  where f⁽ⁿ⁾(a) is the n-th derivative of f evaluated at a. Calculating the derivatives can be challenging, but it's a systematic way to find the power series representation.

• Laurent Series: Laurent series are a generalization of Taylor series that allow for negative powers of (x - a). They are used to represent functions that have singularities (points where they are not analytic).

• Complex Analysis: In complex analysis, power series play a fundamental role in defining analytic functions. The theory of complex power series is very rich and provides powerful tools for analyzing complex functions.

Conclusion

Identifying the function represented by a power series is a fundamental skill in mathematical analysis. By mastering the techniques of pattern recognition, manipulation, and comparison with known series, you can unlock the power of power series to represent and analyze a wide range of functions. Remember to always consider the interval of convergence, and be careful to avoid common mistakes. With practice, you'll become proficient at recognizing and working with these powerful mathematical tools.