Problem 28 - Essential Calculus

Question

In Exercises \(7-28,\) find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) $$ \sum_{n=1}^{\infty} \frac{n !(x+1)^{n}}{1 \cdot 3 \cdot 5 \cdots(2 n-1)} $$

Step-by-Step Solution

Answer

The interval of convergence will be the set of all \(x\) values for which the limit from the Ratio Test is less than 1, including any endpoints where series converges.

Step 1: Apply the Ratio Test

The Ratio Test states that a series \(\sum_{n=1}^{\infty} a_n\) converges if \(\lim_{n\to \infty}|a_{n+1}/a_n| < 1\). First, compute the ratio \(|a_{n+1}/a_n|\) for the given power series.

Step 2: Simplify the Ratio Result

Simplify the resulting expression from Step 1 and then take the limit as \(n\) goes to infinity.

Step 3: Find the Interval of Convergence

After finding the limit, set it less than 1 to find the interval of convergence.

Step 4: Check the Endpoints

To be sure about the interval of convergence, check if the series converges at the endpoints of the interval found in Step 3.

Key Concepts

Interval of Convergence

The interval of convergence is the set of all values of \(x\) for which a power series converges. For a power series centered at \(x = a\), it often initially appears as an interval \((a - R, a + R)\), where \(R\) is the radius of convergence. Identifying this interval gives us a clear understanding of where the series behaves nicely and converges to a finite value.

• Find this interval by setting the limit of the simplified ratio (from the Ratio Test) less than 1.
• Always remember to verify the validity of this interval by checking the endpoints separately.

This approach ensures that we correctly identify all \(x\) values for which the series converges.

Ratio Test

The Ratio Test is a powerful tool for determining the convergence of series. It simplifies the often-complex task of convergence checking by focusing on the ratio of successive terms. For a series \(\sum a_n\), if \(\lim_{n \to \infty} |a_{n+1}/a_n| < 1\), the series converges.

• Apply this by calculating \(|a_{n+1}/a_n|\) using the terms of your series.
• Simplify the expression, and then take the limit as \(n\) approaches infinity.
• If the limit is less than one, your series converges on that interval.

This method provides a reliable way to find the radius of convergence.

Convergence

Convergence in the context of power series refers to finding whether the infinite sum approaches a finite value. Achieving convergence is essential for the series to be useful in applications.

• A series converges if the sum of its terms tends towards a single number as more and more terms are added.
• Using tests like the Ratio Test helps determine convergence over different intervals.

Paying attention to whether the series converges is fundamental before using it for calculations or approximations.

Endpoints

Endpoints are the critical parts of the interval of convergence where special attention is needed. These boundary points require individual consideration to ensure accurate results.

• After determining the initial interval using the Ratio Test, always test the endpoints separately.
• Plug the endpoints into the original series to see if the resulting series converges.
• This step keeps your interval accurate and reliable.

Double-checking endpoints ensures that every possible case for \(x\) is covered in your analysis.