Problem 14 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 9-28, find the convergence set for the given power series. Hint: First find a formula for the nth term; then use the Absolute Ratio Test. $$ x+2^{2} x^{2}+3^{2} x^{3}+4^{2} x^{4}+\cdots $$

Step-by-Step Solution

Answer

The convergence set is \((-1, 1)\).

Step 1: Identify the nth Term

The given series is \( x + 2^{2} x^{2} + 3^{2} x^{3} + 4^{2} x^{4} + \cdots \). Notice the pattern in each term: it is \( n^{2} x^{n} \). Therefore, we can express the nth term of the series as \( a_n = n^2 x^n \).

Step 2: Set the Absolute Ratio

To find the convergence set, we will use the Absolute Ratio Test. This requires the ratio \( \left| \frac{a_{n+1}}{a_n} \right| \). Calculate the ratio: \[ \frac{a_{n+1}}{a_n} = \frac{(n+1)^2 x^{n+1}}{n^2 x^n} = \frac{(n+1)^2}{n^2} \cdot x \] Simplifying gives \[ \frac{a_{n+1}}{a_n} = \left( \frac{n+1}{n} \right)^2 \cdot x \].

Step 3: Apply the Limit for Convergence

According to the Absolute Ratio Test, a series converges absolutely if the limit \[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 \]. Evaluate the limit:\[ \lim_{n \to \infty} \left| \frac{(n+1)^2}{n^2} \cdot x \right| = \lim_{n \to \infty} \left| \left( 1 + \frac{1}{n} \right)^2 \right| \cdot |x| = |x| \], because \( \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^2 = 1 \).

Step 4: Determine the Convergence Set

Set the result of the limit from Step 3 to be less than 1 for convergence: \[ |x| < 1 \].The series converges when \( |x| < 1 \). This means it converges for all \( x \) within the interval \( (-1, 1) \).

Key Concepts

absolute ratio test

The Absolute Ratio Test is an important method for determining the convergence of a series. It focuses on the limit of the ratio of successive terms in the series.
For a series with terms denoted as \( a_n \), the test examines 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, meaning it converges regardless of whether the terms are positive or negative. In the context of power series, such as \( x + 2^2 x^2 + 3^2 x^3 + \ldots \), the Absolute Ratio Test aids in figuring out where the series converges on the real number line.
By applying this test, we calculate the ratio between successive series terms and then take their limit as \( n \) approaches infinity. If the outcome of that limit is less than one, the original series converges. This process not only confirms convergence but also helps in identifying the range of \( x \) values where the series is valid.

nth term

The nth term of a series is crucial for understanding and manipulating series convergence. It essentially defines the general term of the series in consideration and can be used for various tests, including the Absolute Ratio Test.
In our given series, which looks like:

• \( x + 2^2 x^2 + 3^2 x^3 + 4^2 x^4 + \cdots \),

the nth term is described by the formula \( a_n = n^2 x^n \). Recognizing this term helps identify the pattern within the series, which informs subsequent analyses like finding convergence intervals.
Writing the series in terms of its nth term provides a clearer view of how the series behaves as it progresses. This is because each term includes both the variable \( x \) raised to a power and a coefficient in the form of \( n^2 \). Knowing the nth term lets us easily manipulate and test the series for convergence by applying methods such as the Absolute Ratio Test or others.

convergence intervals

Convergence intervals are the ranges of values for which a power series converges. For any power series, knowing these intervals is key to understanding where the series is valid.
In this case, our series

• \( x + 2^2 x^2 + 3^2 x^3 + \ldots \)

converges on the interval \( (-1, 1) \).
To determine these intervals, one typically employs convergence tests like the Absolute Ratio Test. For this series, using the Absolute Ratio Test showed that

• \( |x| < 1 \) for convergence

This results in the interval \( (-1, 1) \), meaning the series sums to a finite value whenever \( x \) is between \(-1\) and \(1\). Such understanding is essential in fields like real analysis and complex analysis, where knowing these intervals allows for more precise application of series in solving mathematical problems.
Overall, convergence intervals provide insight on the domain where the power series retains its validity and practicality.