Problem 25 - Calculus: Early Transcendentals

Question

Evaluate the indefinite integral as a power series. What is the radius of convergence? \( \int \frac {t}{1 - t^8} dt \)

Step-by-Step Solution

Answer

The power series is \( \sum_{n=0}^{\infty} \frac{t^{8n+2}}{8n+2} + C \) with a radius of convergence of 1.

Step 1: Express Integrand as a Geometric Series

First, recognize that the function \( \frac{1}{1-t^8} \) can be expanded as a geometric series. Recall that \( \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n \) for \(|x| < 1\). In this case, replace \( x \) with \( t^8 \), giving us: \[ \frac{1}{1-t^8} = \sum_{n=0}^{\infty} (t^8)^n = \sum_{n=0}^{\infty} t^{8n} \] This series is valid when \( |t^8| < 1 \) or \( |t| < 1 \).

Step 2: Form Integral as a Series

Substitute the series back into the given integral: \[ \int \frac{t}{1-t^8} dt = \int t \left( \sum_{n=0}^{\infty} t^{8n} \right) dt \] This can be written as: \[ \int \sum_{n=0}^{\infty} t^{8n+1} dt \]

Step 3: Integrate Term by Term

We now integrate each term \( t^{8n+1} \) term by term: \[ \int t^{8n+1} dt = \frac{t^{8n+2}}{8n+2} \] Thus, the integral becomes: \[ \sum_{n=0}^{\infty} \frac{t^{8n+2}}{8n+2} + C \] where \( C \) is the constant of integration.

Step 4: Determine Radius of Convergence

Since the power series expansion of \( \frac{1}{1-t^8} \) is valid for \( |t| < 1 \), the radius of convergence of the power series \( \sum_{n=0}^{\infty} \frac{t^{8n+2}}{8n+2} \) is also the same region. Therefore, the radius of convergence is 1.

Key Concepts

Power Series

A power series is a representation of a function as an infinite sum of terms involving powers of a variable. In this scenario, we use the concept of power series to simplify the integration process. When integrating functions that resemble the standard form \( \frac{1}{1-x} \), we expand them into a power series for easier integration.

• Each term in the series is typically expressed as \( x^n \), where \( n \) begins at zero and goes to infinity.
• A key characteristic of a power series is its ability to approximate complex functions with high accuracy within its interval of convergence.

For the integrand \( \frac{t}{1-t^8} \), we substitute \( t^8 \) in place of \( x \) in the geometric series. This conversion employs a power series to expand the function inside the integral, allowing for an easier term-by-term integration. Power series are instrumental in calculus for providing precise estimates and solving integrals that would otherwise be complex.

Radius of Convergence

The radius of convergence is a critical concept when working with power series. It indicates the interval within which the series converges to a specific value, meaning the series will produce accurate results within this domain.In the context of integration, determining the radius of convergence is vital to know the domain over which our integrated series is valid.

• The series \( \sum_{n=0}^{\infty} (t^8)^n \) is based on the geometric series \( \frac{1}{1-t^8} \), converging for \( |t^8| < 1 \), simplifying to \( |t| < 1 \).
• Thus, the overall integral expressed as a power series converges in the same interval, giving it a radius of convergence of 1.

In practice, understanding the radius of convergence helps us validate where the resultant power series solution is applicable, maintaining mathematical accuracy in computations.

Geometric Series

A geometric series is a simple, yet powerful tool in mathematics, used to sum a sequence of numbers where each term after the first is found by multiplying the previous one by a constant, referred to as the common ratio. The expression \( \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n \) is the standard form of a geometric series for \(|x| < 1\) and is quite useful in calculus.

• By recognizing the form \( \frac{1}{1-t^8} \), we rewrite it as a geometric series \( \sum_{n=0}^{\infty} (t^8)^n \).
• This allows us to integrate the function by focusing on each term of the series separately.

Geometric series simplify the process of integration, allowing for easy term-by-term integration, especially when dealing with functions that fit or can be manipulated into the geometric series form. This flexibility makes them particularly useful across various mathematical problems, including those involving series and integrals.