Problem 24 - Calculus: Early Transcendentals

Question

Find the radius of convergence and interval of convergence of the series. \( \sum_{n = 1}^{\infty} \frac {n^2x^n}{2 \cdot 4 \cdot 6 \cdot \cdot \cdot \cdot \cdot (2n)} \)

Step-by-Step Solution

Answer

Radius of convergence: 2; Interval of convergence: \((-2, 2)\).

Step 1: Identify the General Term

The general term of the series is \( a_n = \frac{n^2 x^n}{2 \cdot 4 \cdot 6 \cdot \cdot \cdot \cdot (2n)} \). Notice that the denominator is the product of the first \( n \) even numbers, which is equal to \( 2^n \cdot n! \). Thus, the general term can be rewritten as \( a_n = \frac{n^2 x^n}{2^n \cdot n!} \).

Step 2: Apply the Ratio Test

To find the radius of convergence, apply the Ratio Test: \[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{(n+1)^2 x^{n+1}}{2^{n+1} (n+1)!} \cdot \frac{2^n n!}{n^2 x^n} \right|. \]

Step 3: Simplify the Ratio

Simplify the ratio:\[ \lim_{n \to \infty} \left| \frac{(n+1)^2 x}{2(n+1) n^2} \right| = \lim_{n \to \infty} \left| \frac{(n+1)x}{2n^2} \right| \cdot \left| \frac{n+1}{n} \right| \approx \left| \frac{x}{2} \right| \cdot 1 = \frac{|x|}{2}. \]

Step 4: Determine the Radius of Convergence

According to the Ratio Test, the series converges when \( \frac{|x|}{2} < 1 \). Solving for \( x \), we derive that the radius of convergence, \( R \), is \( 2 \). Thus, \( |x| < 2 \).

Step 5: Determine the Interval of Convergence

Test the endpoints \( x = -2 \) and \( x = 2 \) to find the interval of convergence.- For \( x = 2 \), the series becomes \( \sum_{n=1}^{\infty} \frac{n^2 2^n}{2^n \cdot n!} = \sum_{n=1}^{\infty} \frac{n^2}{n!} \), which diverges as it grows faster than geometric.- For \( x = -2 \), the series becomes \( \sum_{n=1}^{\infty} \frac{n^2 (-2)^n}{(2^n \cdot n!)} \), which also diverges due to the factor of \((-1)^n\), as in a series like \( \sum_{n=1}^{\infty} \frac{n^2 (-1)^n}{n!} \).

Step 6: Conclusion on the Interval

Since both endpoint tests result in divergence, the interval of convergence is \((-2, 2)\). The endpoints are not included.

Key Concepts

Interval of Convergence

The interval of convergence tells us where a power series converges on the real number line. For the given series, we determined that the interval of convergence is (-2, 2). This means that for every value of \( x \) within this interval, the series converges. However, it diverges at \( x = -2 \) and \( x = 2 \), so these points are not included in our interval.
To test the endpoints, we substitute \( x = -2 \) and \( x = 2 \) into the series, and both cases result in divergence, confirming our findings.

• This process helps confirm where exactly the series transitions from converging to diverging.
• Such analysis is crucial for applying the series within certain bounds while ensuring accuracy.

Ratio Test

The ratio test is a powerful tool in determining the radius of convergence for a series. We use it to assess the behavior of the series as \( n \) tends to infinity. In our exercise, the test involves evaluating
\[\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\]Where \( a_n \) is the general term of the series. If this limit is less than 1, the series converges. If it's greater than 1, the series diverges. Precisely, setting \( \frac{|x|}{2} < 1 \) allows us to solve for \( x \), which is essential in finding the radius of convergence. In this manner, by calculating we found a radius of 2, leading us to the interval \(|x| < 2\).

• It simplifies evaluating power series to determine convergence or divergence.
• Useful as it applies to a vast range of series, making it widely applicable in analysis.

Power Series

A power series is a series of the form \( \sum_{n=0}^{\infty} c_n(x-a)^n \), where \( c_n \) are coefficients and \( a \) is the center. In this problem, the series uses a general term \( a_n \), indicating coefficients related to \( x^n \). Through convergence testing, these series allow us to model functions within specific intervals, helping approximate complex functions.
The flexibility of power series is instrumental for effortlessly solving differential equations and evaluating functions within their convergence interval.

• Power series foster an iterative approach that can adapt to diverse conditions.
• They ensure precision when a function is too intricate to express using elementary functions.

Convergence Testing

Convergence testing is integral to understanding series behaviors and plays a crucial role in mathematical analysis. Through different tests, including the ratio test, we determine whether the series converges or diverges for specific values.
This exercise focused on identifying these behaviors for a particular power series, highlighting the importance of applying various tests to ensure complete analysis.
The ratio test gives an initial scope of convergence through simplifying and evaluating the limits of series terms, supplemented by endpoint testing for a conclusive interval.

• It's essential for ensuring that series approximations are valid and accurate.
• These methods ensure the series are functional within determined bounds, avoiding errors in mathematical applications.