Problem 10 - Calculus Early Transcendentals: Pearson New International Edition

Question

find the power series representation for \(f(x)\) and specify the radius of convergence. Each is somehow related to a geometric series. $$ f(x)=\int_{0}^{x} \tan ^{-1} t d t $$

Step-by-Step Solution

Answer

The power series for \( f(x) \) is \( \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+2}}{(2n+1)(2n+2)} \) with radius of convergence 1.

Step 1: Understand the Integral Representation

The function given is an integral of the arctangent function, \( f(x) = \int_{0}^{x} \tan^{-1} t \, dt \). To find the power series, first, we should find the power series representation for \( \tan^{-1} t \).

Step 2: Power Series for \( \tan^{-1} t \)

The function \( \tan^{-1} t \) can be represented as a power series: \( \tan^{-1} t = \sum_{n=0}^{\infty} (-1)^n \frac{t^{2n+1}}{2n+1} \). This series is derived from the inverse tangent function.

Step 3: Integrate the Power Series Term-by-Term

Since \( f(x) = \int_{0}^{x} \tan^{-1} t \, dt \), replace \( \tan^{-1} t \) with its power series representation to get \( f(x) = \int_{0}^{x} \sum_{n=0}^{\infty} (-1)^n \frac{t^{2n+1}}{2n+1} \, dt \). Integrate term-by-term: \( f(x) = \sum_{n=0}^{\infty} (-1)^n \frac{1}{2n+1} \left[ \frac{t^{2n+2}}{2n+2} \right]_{0}^{x} \).

Step 4: Simplify the Expression

After integration, substitute the limits: \( f(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+2}}{(2n+1)(2n+2)} \). This expression represents the series for the function \( f(x) \).

Step 5: Determine the Radius of Convergence

The radius of convergence for the series \( \tan^{-1} t \) is found by using the formula for a standard geometric series, which is convergent for \(|t| < 1\). Therefore, this dictates that the power series for \( f(x) \) is convergent for \(|x| < 1\), with a radius of convergence of 1.

Key Concepts

Geometric Series

A geometric series is a type of series where each term is a fixed multiple, called the common ratio, of the previous term. In mathematical terms, a geometric series can be written as: \[ a + ar + ar^2 + ar^3 + \dots \]where \(a\) is the first term and \(r\) is the common ratio. A key characteristic of geometric series is that they can converge only when the absolute value of the common ratio \(|r|\) is less than 1. This makes them an important tool in calculus for representing functions, as they have a straightforward convergence criterion and a simple formula for finding their sum when they converge.

In the context of power series, which are a form of infinite series using powers of a variable, geometric series principles are often used to determine convergence. Specifically, they help in identifying the interval within which a power series converges. A power series can often be seen as an extension or transformation of a geometric series, making understanding geometric series foundational to grasping power series behaviors.

Radius of Convergence

The radius of convergence is a crucial concept when dealing with power series. It indicates the range of values over which a power series converges.

For a given power series, the radius of convergence is calculated using the ratio test or root test, which involves examining the limit behavior of the coefficients and terms within the series. For a series:

• \( \sum_{n=0}^{\infty} a_n(x - c)^n \)

the radius of convergence \(R\) can be determined by finding the limit:

• \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \)

Once \(R\) is identified, the series converges when \(|x - c| < R\).

In the exercise, the series derived from integrating \(\tan^{-1} t\) is applicable within the context of its radius of convergence. For \(\tan^{-1} t\), the series converges for \(|t| < 1\), translating to the power series for \(f(x)\) also having a radius of convergence of 1. This means the function converges for \(|x| < 1\). Understanding this concept guides us to know when the series representation of a function gives valid results.

Integrating Power Series

Integrating power series involves taking the indefinite integral of each term within the series. This technique is particularly useful for finding power series representations of integrals that are not easily solvable by standard integration methods.

When integrating a power series term-by-term:

• Start with a series representation of the function you wish to integrate.
• Integrate each term of the series with respect to the variable, adjusting the exponents accordingly.
• Apply the limits of integration if it is a definite integral.

For example, when integrating \( \tan^{-1} t \), its power series representation \( \sum_{n=0}^{\infty} (-1)^n \frac{t^{2n+1}}{2n+1} \) is used. You integrate each term to transform it into the form needed to represent the integral as a series:

\[ \int \sum_{n=0}^{\infty} (-1)^n \frac{t^{2n+1}}{2n+1} \, dt = \sum_{n=0}^{\infty} (-1)^n \frac{1}{2n+1} \left[ \frac{t^{2n+2}}{2n+2} \right]_{0}^{x} \]Using this process, we obtain a new power series that represents the integral, indicating that power series can be a versatile tool in handling integrals and providing solutions where direct integration might be complex or not possible.