Problem 9 - Calculus: Early Transcendentals

Question

Find the radius of convergence and interval of convergence of the series. \( \sum_{n = 1}^{\infty} \frac {x^n}{n^44^n} \)

Step-by-Step Solution

Answer

Radius of convergence is 4, interval of convergence is [-4, 4].

Step 1: Identify the Series

The given series is \( \sum_{n = 1}^{\infty} \frac {x^n}{n^4 4^n} \). To find the radius and interval of convergence, we'll begin with applying the ratio test.

Step 2: Apply the Ratio Test

The ratio test states that for a series \( \sum a_n \), the series converges absolutely if \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 \). Here, \( a_n = \frac{x^n}{n^4 4^n} \). We calculate \( a_{n+1} = \frac{x^{n+1}}{(n+1)^4 4^{n+1}} \).

Step 3: Calculate the Ratio

Substituting \( a_n \) and \( a_{n+1} \) into the ratio, we have:\[\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{x^{n+1} (n^4) 4^n}{x^n (n+1)^4 4^{n+1}} \right| = \left| \frac{x}{4} \cdot \frac{n^4}{(n+1)^4} \right|\]

Step 4: Evaluate the Limit

Taking the limit as \( n \to \infty \), we focus mostly on \( \frac{n^4}{(n+1)^4} \) which tends to 1 as \( n \) becomes very large. We have:\[\lim_{n \to \infty} \left| \frac{x}{4} \cdot \frac{n^4}{(n+1)^4} \right| = \left| \frac{x}{4} \right|\] Thus, by the ratio test, we want \( \left| \frac{x}{4} \right| < 1 \), or \( |x| < 4 \).

Step 5: Determine the Radius of Convergence

From the inequality \( |x| < 4 \), we deduce that the radius of convergence \( R \) is 4.

Step 6: Find the Interval of Convergence

The basic interval of convergence is \( -4 < x < 4 \). We need to check endpoints \( x = -4 \) and \( x = 4 \):1. For \( x = 4 \), the series becomes \( \sum \frac{4^n}{n^4 4^n} = \sum \frac{1}{n^4} \), which converges by the p-series test (p > 1).2. For \( x = -4 \), the series becomes \( \sum \frac{(-4)^n}{n^4 4^n} = \sum (-1)^n \frac{1}{n^4} \), which converges by the alternating series test.Thus, the interval of convergence is \( [-4, 4] \).

Key Concepts

Interval of Convergence

The interval of convergence for a series is the set of all values of the variable that make the series converge. For the series in question, \(\sum_{n=1}^{\infty} \frac{x^n}{n^4 4^n}\), we start by using the Ratio Test to determine where the series converges. The inequality \(|x| < 4\) is derived, indicating that within this boundary, the series converges. But we must also check the endpoints, which are often crucial.- At \(x = 4\), we have the series \(\sum \frac{1}{n^4}\). This is a p-series with \(p = 4\), known to converge because \(p > 1\).- At \(x = -4\), we have \(\sum (-1)^n \frac{1}{n^4}\). The alternating factor \((-1)^n\) turns this into an alternating series, which converges according to the alternating series test because the terms \(\frac{1}{n^4}\) decrease to zero.Thus, the interval of convergence for this series, after including endpoints, is \([-4, 4]\).

Ratio Test

The Ratio Test is a tool used to determine whether a series converges. It works well for series involving products and powers. The test involves examining the limit:\[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \]- If this limit is less than 1, the series converges absolutely.- If it is more than 1, the series diverges.- If it is equal to 1, the test is inconclusive. For the series \(\sum \frac{x^n}{n^4 4^n}\), we calculate:\[ \left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{x}{4} \cdot \frac{n^4}{(n+1)^4} \right| \]As \(n\) approaches infinity, \(\frac{n^4}{(n+1)^4} \to 1\), focusing our limit primarily on \(\left| \frac{x}{4} \right|\). This results in the condition \(|x| < 4\), providing the series with a convergence range based on the test.

Alternating Series Test

The Alternating Series Test helps understand the convergence of series that switch signs. An alternating series is typically expressed as \(\sum (-1)^n a_n\), where \(a_n > 0\).The test states such a series converges if:- The terms \(a_n\) decrease over time.- \(\lim_{n \to \infty} a_n = 0\).For instance, the series resulting from \(x = -4\) becomes \(\sum (-1)^n \frac{1}{n^4}\). Given:- \(a_n = \frac{1}{n^4}\), the sequence clearly decreases.- The limit is zero as \(n\) grows, due to the power \(4\), satisfying the condition.Thus, the series converges by this test, illustrating its importance when determining series behaviors at specific points.

P-Series Test

The p-series test is applied to series of the form \(\sum \frac{1}{n^p}\). The test dictates that:- The series converges if \(p > 1\).- The series diverges if \(p \leq 1\).For the case \(x = 4\), the series simplifies to \(\sum \frac{1}{n^4}\). Here, \(p = 4\), clearly greater than 1, indicating convergence.This specific characteristic makes the p-series test extremely valuable when identifying whether a power series converges at certain boundaries or specific points within its domain. Thus, using a p-series test is a quick and reliable way to assess convergence.