Problem 17 - Essential Calculus Early Transcendentals

Question

Find the radius of convergence and interval of convergence of the series. $$\sum_{n=1}^{\infty} \frac{n}{b^{n}}(x-a)^{n}, \quad b>0$$

Step-by-Step Solution

Answer

The radius of convergence is \( b \). The interval of convergence is \((a-b, a+b)\).

Step 1: Identify the Type of Series

The given series \( \sum_{n=1}^{\infty} \frac{n}{b^{n}}(x-a)^{n} \) is a power series of the form \( \sum_{n=1}^{\infty} c_n (x-a)^n \) where \( c_n = \frac{n}{b^{n}} \). We need to find the radius of convergence using the ratio test.

Step 2: Apply the Ratio Test

For the ratio test, we consider the limit:\[L = \lim_{n \to \infty} \left| \frac{c_{n+1}}{c_n} \right|\]Substituting \( c_{n+1} = \frac{n+1}{b^{n+1}} \) and \( c_n = \frac{n}{b^n} \) into the formula gives:\[L = \lim_{n \to \infty} \left| \frac{\frac{n+1}{b^{n+1}}}{\frac{n}{b^n}} \right| = \lim_{n \to \infty} \frac{n+1}{n} \cdot \frac{1}{b} = \frac{1}{b} \cdot \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)\]This simplifies to \( L = \frac{1}{b} \) as \( n \to \infty \).

Step 3: Find the Radius of Convergence

According to the ratio test, the series converges if \( L < 1 \). Therefore, the series converges when:\[\frac{|x-a|}{b} < 1\]This implies:\[ |x-a| < b \]The radius of convergence \( R \) is the distance from \( a \) to the endpoints of the interval of convergence and is directly given by \( R = b \).

Step 4: Determine the Interval of Convergence

Knowing that \( |x-a| < b \), the interval of convergence is:\[a-b < x < a+b\]To determine the convergence at the end points (\( x = a-b \) and \( x = a+b \)), we would need to check each case separately using an appropriate testing method, such as the alternating series test or direct comparison, if needed.

Key Concepts

Power Series

A power series is an infinite series that includes terms in the form \((x-a)^n\) multiplied by coefficients. This setup creates a function depending on \(x\) which approaches a limit as more terms are added. A general power series about a central value \(a\) is written as:\[ \sum_{n=0}^{fty} c_n (x-a)^n \]where each \(c_n\) represents the coefficients of the series.
In our original problem, we look at a power series given by:\[ \sum_{n=1}^{\infty} \frac{n}{b^{n}} (x-a)^n \]This series has special characteristics due to the coefficients \(\frac{n}{b^n}\). Each term in the series involves \((x-a)\), which shifts the series along the x-axis to center it around the point \(a\).

Power series like this one are significant because they can represent many types of functions, often simplifying complex relationships into manageable series. Understanding the behavior of power series, particularly how they converge, is essential in calculus and mathematical analysis.

Interval of Convergence

The interval of convergence of a power series is the range of \(x\) values for which the series converges to a finite value. Discovering this interval helps understand for which input values your series behaves nicely and sums to a determinable number.

To find the interval of convergence, you start with the radius of convergence \(R\), which is half the length of this interval. For the series \( \sum_{n=1}^{\infty} \frac{n}{b^{n}}(x-a)^{n}\), we determined earlier that the radius of convergence is \(R = b\). This tells us that the series converges when \(|x-a| < b\).

• Calculate \(|x-a| < b\) to find the basic interval: \(a-b < x < a+b\)
• Explore convergence at the endpoints separately as the series might include or exclude them.
  

Within this range, the series participants converge, meaning they sum to a finite total. This interval may only partially or fully include \(a-b\) and \(a+b\), dependent on further testing.

Ratio Test

The ratio test is a common convergence test for series with positive terms, especially useful for power series. This test considers the limit of the absolute ratio of consecutive terms:
\[ L = \lim_{n \to \infty} \left| \frac{c_{n+1}}{c_n} \right| \]
For our series, \(c_n = \frac{n}{b^n}\), the ratio test examines:\[ L = \lim_{n \to \infty} \left| \frac{\frac{n+1}{b^{n+1}}}{\frac{n}{b^n}} \right| = \lim_{n \to \infty} \frac{n+1}{n} \cdot \frac{1}{b} = \frac{1}{b} \cdot \lim_{n \to \infty} \left(1 + \frac{1}{n}\right) \]
This simplifies to \( L = \frac{1}{b}\) as \(n\) goes to infinity.

• If \(L < 1\), the series converges.
• If \(L > 1\), it diverges.
• If \(L = 1\), the test is inconclusive.

The ratio test thus shows our series converges for:\[ \frac{|x-a|}{b} < 1\]leading to our convergence radius \( R = b \), confirming the series converges within the interval \(a-b < x < a+b\). This test gives insight into where the series behaves well and underscored the mathematical interest in convergence behavior.