Problem 18 - 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? $$ \sum_{n=0}^{\infty} \frac{n x^{n}}{4^{n}\left(n^{2}+1\right)} $$

Step-by-Step Solution

Answer

The series converges absolutely on \((-4, 4)\) and conditionally nowhere.

Step 1: Recognize the series

The given series is \( \sum_{n=0}^{\infty} \frac{n x^{n}}{4^{n}\left(n^{2}+1\right)} \). We need to find the radius and interval of convergence and determine where it converges absolutely and conditionally.

Step 2: Apply the Ratio Test

To determine the radius of convergence, we use the Ratio Test. Calculate \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_{n}} \right| \) where \( a_n = \frac{n x^{n}}{4^{n}(n^{2}+1)} \).

Step 3: Simplify and compute the limit

Compute \(\lim_{n \to \infty} \left| \frac{(n+1)x^{n+1}}{4^{n+1}((n+1)^2+1)} \cdot \frac{4^n(n^2+1)}{n x^n} \right| \). Simplifying, we get \( |x| \cdot \frac{1}{4} \cdot \lim_{n \to \infty} \frac{n+1}{n} \cdot \frac{n^2+1}{(n+1)^2+1} \).

Step 4: Evaluate the limit

The limit becomes \(|x| \cdot \frac{1}{4} \cdot \lim_{n \to \infty} \left( \frac{n^2+1}{(n+1)^2+1} \right)\). Simplifying this further results in \(|x| \cdot \frac{1}{4} \cdot 1 = \frac{|x|}{4}\) as \(n \to \infty \).

Step 5: Find the Radius of Convergence

Set \( \frac{|x|}{4} < 1 \) to ensure convergence. Thus, \( |x| < 4 \). Therefore, the radius of convergence, \( R \), is \( 4 \).

Step 6: Determine the Interval of Convergence

Using the radius, the interval before testing endpoints is \((-4, 4)\). Evaluate convergence at the endpoints \( x = -4 \) and \( x = 4 \) using the original series.

Step 7: Test Convergence at \(x = 4\)

Substitute \( x = 4 \) into the series to get \( \sum_{n=0}^{\infty} \frac{n}{n^{2}+1} \), which diverges by the limit comparison test with \( \sum \frac{1}{n} \).

Step 8: Test Convergence at \(x = -4\)

Substitute \( x = -4 \) into the series to get \( \sum_{n=0}^{\infty} \frac{(-1)^n n}{n^{2}+1} \), which also diverges since \( \sum \frac{n}{n^{2}+1} \sim \sum \frac{1}{n} \).

Step 9: Conclude Interval of Convergence

Since the series diverges at both endpoints \( x = 4 \) and \( x = -4 \), the interval of convergence is \((-4, 4)\).

Step 10: Identify Absolute Convergence

Within \((-4, 4)\), the series converges absolutely wherever \( \frac{|x|}{4} < 1 \) holds true, which is over the entire interval \((-4, 4)\).

Step 11: Identify Conditional Convergence

Since the series diverges at \( x = -4 \) and \( x = 4 \), there are no points of conditional convergence outside \((-4, 4)\).

Key Concepts

Interval of Convergence

To determine where a series converges, we need to analyze its interval of convergence. This term refers to the range of values of the variable, usually denoted by \( x \), for which the series converges to a finite value. The interval of convergence is typically discovered using the radius of convergence, which is found via the Ratio Test. Once the radius is known, we can express the interval of convergence.
In our exercise, the series \( \sum_{n=0}^{\infty} \frac{n x^n}{4^n\left(n^2+1\right)} \) has a radius of convergence \( R = 4 \). This means the series converges for \( |x| < 4 \), so the potential interval before testing endpoints is \((-4, 4)\).
Next, testing the endpoints \( x = -4 \) and \( x = 4 \) shows divergence at both points, hence they are not included in the interval. The series converges on the open interval \((-4, 4)\), ensuring all values within this interval satisfy the series' convergence conditions.
Evaluating the endpoints is key because, without it, the series might mistakenly be assumed to converge or diverge where it does the opposite.

Absolute Convergence

Absolute convergence is a stronger form of convergence where the positive term series also converges. This form of convergence implies that the series will still converge even if we replace all terms with their absolute values.
In this exercise, within the interval \((-4, 4)\), the series converges absolutely. This is determined using the Ratio Test result \( \frac{|x|}{4} < 1 \). In simpler terms, if each term in the series is replaced by its absolute value, the series remains convergent over all \( x \) where \( |x| < 4 \).
Understanding absolute convergence is critical as it assures that potential rearrangements of the series won't affect its convergence, which is not always the case with conditional convergence.

Conditional Convergence

Conditional convergence is a more nuanced situation where a series converges, but does not converge absolutely. This means if you took the absolute value of each term in the series, that new series would diverge.
In our series, there are no cases of conditional convergence within or on the boundary of the interval \((-4, 4)\). The series either absolutely converges or diverges within the interval or at its endpoints.
Understanding conditional convergence helps especially with handling functions having alternating terms, but here, the series does not exhibit conditional convergence because it doesn't converge at the endpoints.

Ratio Test

The Ratio Test is central to determining the radius and interval of convergence for series. This test involves taking the limit of the absolute value of the ratio of consecutive terms of the series.
For our series, if we let \( a_n = \frac{n x^n}{4^n(n^2+1)} \), the Ratio Test findings were obtained by evaluating \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_{n}} \right| \). After simplification, it reduced to \( \frac{|x|}{4} \).
According to the test, for convergence, we need \( \frac{|x|}{4} < 1 \). Solving this, we get \( |x| < 4 \), hence the radius is 4.
The Ratio Test is powerful because it simplifies finding convergence criteria, especially in power series, making it easier to find where the series converges absolutely or conditionally.