Problem 31 - Essential Calculus Early Transcendentals

Question

A function \(f\) is defined by $$f(x)=1+2 x+x^{2}+2 x^{3}+x^{4}+\cdots$$ that is, its coefficients are \(c_{2 n}=1\) and \(c_{2 n+1}=2\) for all \(n \geqslant 0 .\) Find the interval of convergence of the series and find an explicit formula for \(f(x)\)

Step-by-Step Solution

Answer

The interval of convergence is \(-1 < x < 1\) and the function is \(f(x) = \frac{1 + 2x}{1 - x^2}.\)

Step 1: Identify the Function Type

The function given is an infinite series that can be rewritten as \( f(x) = \sum_{n=0}^{ ightarrow} (c_{2n}x^{2n} + c_{2n+1} x^{2n+1}) \). The coefficients \(c_{2n} = 1\) for even powers and \(c_{2n+1} = 2\) for odd powers. To analyze this better, note the alternating coefficients in the series.

Step 2: Separate the Series into Two Parts

Split the series into two separate sums based on the parity of the power: \[f(x) = \sum_{n=0}^{ ightarrow} x^{2n} + \sum_{n=0}^{ ightarrow} 2x^{2n+1}\].This splits the series into terms involving even powers and odd powers of \(x\).

Step 3: Calculate the Sum of the Even Power Series

The series \(\sum_{n=0}^{ ightarrow} x^{2n}\) is a geometric series where \(a=1\) and \(r=x^2\). The sum of this geometric series is \[\sum_{n=0}^{ ightarrow} x^{2n} = \frac{1}{1-x^2}, \text{ for } |x^2|<1 \text{ or } |x|<1.\].

Step 4: Calculate the Sum of the Odd Power Series

The series \( \sum_{n=0}^{ ightarrow} 2x^{2n+1} = 2x(\sum_{n=0}^{ ightarrow} x^{2n}) \). Using the result of Step 3 for the series, this becomes\[\sum_{n=0}^{ ightarrow} 2x^{2n+1} = \frac{2x}{1-x^2}.\].

Step 5: Combine the Series Sums

Add the results from Step 3 and Step 4 to get the overall function:\[f(x) = \frac{1}{1-x^2} + \frac{2x}{1-x^2} = \frac{1 + 2x}{1 - x^2}.\].

Step 6: Determine the Interval of Convergence

The series components in Steps 3 and 4 both converge for \(|x|<1\). Hence, \(f(x)\) converges in the interval \[-1 < x < 1.\].

Key Concepts

Geometric Series

Understanding geometric series is crucial in analyzing infinite series. A geometric series is expressed in the form \( S = a + ar + ar^2 + ar^3 + \cdots \), where \( a \) is the first term and \( r \) is the common ratio. The series has a finite sum if the magnitude of the ratio \( |r| < 1 \).
Key characteristics of a geometric series include:

• The series converges if \( |r| < 1 \), meaning it approaches a single finite number as more terms are added.
• If \( |r| \geq 1 \), the series diverges, which means it does not settle to a finite limit.

For examples, the series \( 1 + x + x^2 + x^3 + \cdots \) is geometric with \( a = 1 \) and \( r = x \). A remarkable property of a converging geometric series is its sum formula: \[S = \frac{a}{1-r} \]This helps to calculate the sum of an infinite series quickly, provided the condition \( |r| < 1 \) is satisfied.

Power Series

A power series is an infinite sum of the form \( \sum_{n=0}^{\infty} c_n (x-a)^n \), where \( c_n \) are coefficients, \( a \) is the center, and \( x \) is the variable. It generalizes polynomials and can extend to functions involving infinitely many terms.
Power series can take the shape of many familiar functions such as exponential, sine, and cosine functions through specific manipulations of their terms. For instance:

• \( e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \)
• \( \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} \)

The primary focus when working with power series is determining the interval over which they converge, allowing us to effectively use them in approximating functions and performing calculus operations such as derivatives and integrals.

Interval of Convergence

The interval of convergence is a critical concept for power series that specifies the values of \( x \) for which the series converges. For a given power series \( \sum_{n=0}^{\infty} c_n (x-a)^n \), understanding convergence is essential in ensuring the series represents a valid function.
To find this interval:

• Determine the radius of convergence \( R \) using methods like the Ratio Test or Root Test.
• The series converges absolutely when \( |x-a| < R \).
• Test endpoints separately because the series might converge conditionally at these points.

For the series in our example, the series is split into two components: an even power series and an odd power series. Both converge when \(|x| < 1\). Thus, the interval of convergence is \(-1 < x < 1\). Surefooting in recognizing and calculating this interval helps ensure that any functions represented by power series remain accurate and useful in this domain.

Explicit Formula

An explicit formula provides a direct expression of a series in a closed form, capturing its entire behavior in a single equation. This negates the need to perform endless additions, as all terms are encapsulated succinctly.
In the given problem, the series for the function \( f(x) = 1 + 2x + x^2 + 2x^3 + \dots \) was split and transformed into geometric series components. Converting these series into closed forms gives:

• Even powers: \( \frac{1}{1-x^2} \)
• Odd powers: \( \frac{2x}{1-x^2} \)

Combined, these produce the explicit formula: \[f(x) = \frac{1 + 2x}{1 - x^2}\]This formula offers a powerful insight, enabling us to easily calculate values of \( f(x) \) within its radius of convergence. Such explicit formulations are indispensable tools in mathematics, granting succinct and functional interpretations of infinite series.