Problem 24 - Thomas Calculus

Question

In Exercises \(1-36\) , (a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$ \sum_{n=1}^{\infty}(\ln n) x^{n} $$

Step-by-Step Solution

Answer

(a) Radius is 1; interval: \(-1 < x < 1\). (b) Absolutely for \(-1 < x < 1\). (c) No conditional convergence.

Step 1: Test for Absolute Convergence

To find series convergence, start with the Ratio Test. The general term of the series is \(a_n = (\ln n)x^n\). We calculate:\[\lim_{n\to\infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n\to\infty} \left| \frac{\ln(n+1)\cdot x^{n+1}}{\ln n\cdot x^n} \right| = \lim_{n\to\infty} \left| \frac{\ln(n+1)}{\ln n}\cdot x \right|\]This simplifies to \( |x| \) as \( \ln(n+1)/\ln n \to 1 \). The series converges absolutely when \(|x| < 1\).

Step 2: Find the Radius and Interval of Convergence

According to the Ratio Test, convergence occurs for \( |x| < 1 \). This indicates that the radius of convergence \( R = 1 \). Therefore, the interval of convergence is \(-1 < x < 1\).

Step 3: Check Convergence at the Endpoints

To determine if the series converges at \( x = -1 \) or \( x = 1 \), substitute these values into the series. For \( x = 1 \), the series becomes \( \sum_{n=1}^{\infty} \ln n \), which diverges. For \( x = -1 \), the series \( \sum_{n=1}^{\infty} \ln n (-1)^n \) diverges as it does not approach zero or exhibit alternating convergence.

Step 4: Identify Absolute vs. Conditional Convergence

The series converges absolutely for \( -1 < x < 1 \) due to the Ratio Test, as this interval satisfies the condition \(|x| < 1\). There is no conditional convergence because series diverges at the endpoints.

Key Concepts

Radius of Convergence

When examining a power series, one key factor to consider is the radius of convergence. This term refers to how far, in terms of the variable \(x\), we can move away from the center of the series (often 0) and still have the series converge. For the series given in the problem, \[ \sum_{n=1}^{\infty} (\ln n) x^n, \] we used the Ratio Test to find that the radius of convergence \( R \) is 1. Essentially, the radius tells us that the series forms a convergent sequence of sums whenever the absolute value of \(x\) is less than 1. In simpler terms, the series will behave well and reach a finite value for any \(x\) value inside this radius, i.e., for \( |x| < 1 \). If \(|x| \geq 1\), the series does not converge – meaning it piles up to infinity, or doesn't settle to a single, finite value.

Interval of Convergence

Knowing the radius of convergence gives us a starting point, but to fully understand the behavior of a series, we must also determine the interval of convergence. This tells us not only within what bounds a series converges, but also about the behavior at the edges of these bounds. For our specific series, from the calculation, the interval of convergence was found to be from \(-1\) to \(1\). Written as an inequality, this is expressed as \( -1 < x < 1 \).However, we also examined the end points:

• At \( x = 1 \), the series diverges. It grows too fast, so it won't reach a finite limit.
• Similarly, at \( x = -1 \), the series does not converge.

Thus, the series converges within the open interval \(-1 < x < 1\), excluding both endpoints.

Ratio Test

The Ratio Test is a powerful tool that helps us determine when a series converges by taking the limit of the absolute values of the ratio of successive terms. In our exercise, applying the Ratio Test involved \[\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{\ln(n+1) \cdot x^{n+1}}{\ln n \cdot x^n} \right|\]Simplification of this expression gave \[ |x| \] as the natural logarithm ratio approaches 1 for very large \( n \). Hence, the test told us the series converges absolutely when \( |x| < 1 \).

• If the resulting limit is less than 1, the series is absolutely convergent within that range of \( x \).
• If the limit is greater than 1, the series diverges.
• If exactly 1, the test is inconclusive, and further investigation is necessary.

Applying the ratio test is a great way to get a quick sense of the behavior of a series, though it may not tell us everything, such as what happens at the boundaries of convergence without further analysis.