Identify The Function Represented By The Following Power Series

Identifying the Function Represented by a Power Series

Power series are fundamental tools in mathematics, particularly in analysis and calculus. They allow us to represent functions as infinite sums of simpler terms, often polynomials. This representation offers numerous advantages, including easier manipulation, approximation, and solution of differential equations. A crucial skill is identifying the function that a given power series represents. This article explores various techniques and strategies for this task, encompassing numerous examples to solidify understanding.

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. Determining the interval of convergence is a critical first step in identifying the function the series represents.

Methods for Identifying the Function

Several techniques can be employed to identify the function represented by a power series. These include:

1. Recognizing Known Power Series Expansions

The most straightforward approach is to recognize the given power series as the Taylor or Maclaurin series of a known function. Memorizing the expansions of common functions is incredibly valuable. Some key examples include:

• Geometric Series: ∑_n=0^∞ xⁿ = 1/(1 - x), |x| < 1
• Exponential Function: ∑_n=0^∞ xⁿ/n! = e^x, for all x
• Sine Function: ∑_n=0^∞ (-1)ⁿx²ⁿ⁺¹/(2n+1)! = sin(x), for all x
• Cosine Function: ∑_n=0^∞ (-1)ⁿx²ⁿ/(2n)! = cos(x), for all x
• ln(1+x): ∑_n=1^∞ (-1)ⁿ⁺¹xⁿ/n = ln(1+x), -1 < x ≤ 1

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

This series is a geometric series with r = x/2. Therefore, the function is:

f(x) = 1 / (1 - x/2) = 2 / (2 - x), provided |x/2| < 1, which means |x| < 2.

2. Using the Taylor or Maclaurin Series Formula

If the series doesn't immediately resemble a known expansion, the next step is to utilize the Taylor or Maclaurin series formula. The Maclaurin series is a special case of the Taylor series where the center c = 0.

The Taylor series of a function f(x) centered at c is given by:

∑_n=0^∞ f⁽ⁿ⁾(c)(x - c)ⁿ/n!

where f⁽ⁿ⁾(c) is the nth derivative of f(x) evaluated at x = c.

Example 2: Identify the function represented by the power series ∑_n=0^∞ (-1)ⁿx²ⁿ/(2n)!

This resembles the cosine function's Maclaurin series. Let's verify:

f(x) = cos(x) f'(x) = -sin(x) f''(x) = -cos(x) f'''(x) = sin(x) f''''(x) = cos(x)

... and so on. Evaluating these derivatives at x = 0 and substituting into the Maclaurin series formula yields the given power series. Therefore, the function is cos(x).

3. Term-by-Term Differentiation or Integration

Sometimes, a power series can be related to a known series through differentiation or integration. This is particularly useful when dealing with series that resemble the derivatives or integrals of familiar functions.

Example 3: Identify the function represented by ∑_n=1^∞ nx^n-1.

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

d/dx [∑_n=0^∞ xⁿ] = ∑_n=1^∞ nx^n-1 = d/dx [1/(1-x)] = 1/(1-x)².

Therefore, the function is 1/(1-x)² for |x| < 1.

4. Using Partial Fraction Decomposition

If the coefficients of the power series are complex or don't directly match known series, partial fraction decomposition might help simplify the expression. This technique is particularly useful when dealing with rational functions.

Example 4: (A more complex example requiring partial fraction decomposition is best illustrated with a specific example, but it would exceed the space constraints here. The basic idea is to decompose a rational function into simpler fractions whose power series are known or easily determined.)

5. Utilizing the Ratio Test for the Radius of Convergence

The radius of convergence helps define the interval where the power series converges. The ratio test is a common method for finding the radius of convergence. It's often the first step before attempting to identify the function, as it provides crucial information about the function's domain of validity.

Example 5: Find the radius of convergence of ∑_n=0^∞ (xⁿ)/(n!).

Using the ratio test:

lim_n→∞ |a_n+1/a_n| = lim_n→∞ |xⁿ⁺¹/(n+1)!| * |n!/xⁿ| = lim_n→∞ |x|/(n+1) = 0

Since the limit is 0, the series converges for all x (radius of convergence is infinite). This confirms the series represents the exponential function e^x.

Advanced Techniques and Considerations

Identifying the function represented by a power series can be challenging. These more advanced techniques and considerations can be helpful:

• Manipulation of Indices: Sometimes, adjusting the starting index of summation or re-indexing the series can transform it into a recognizable form.
• Substitution: Substituting a new variable can simplify the series and reveal its underlying function.
• Comparison with Known Series: Comparing the series to known expansions (even if not exact matches) can provide clues and insights.
• Computer Algebra Systems (CAS): Software like Mathematica or Maple can be invaluable for finding the function represented by a complex power series.

Conclusion

Identifying the function represented by a given power series is a crucial skill in advanced mathematics. Mastering this skill requires a solid understanding of power series, Taylor and Maclaurin series, and various analytical techniques. By combining knowledge of standard series expansions, skillful manipulation, and a systematic approach, one can effectively determine the function represented by even complex power series. Remember to always determine the radius of convergence to understand the domain where the identified function is valid. The examples provided offer a practical starting point, but continuous practice and exploration of more complex cases will further solidify understanding and expertise.