Problem 82 - Essential Calculus

Question

Find all values of \(x\) for which the series converges. For these values of \(x,\) write the sum of the series as a function of \(x\). $$ \sum_{n=1}^{\infty}\left(\frac{x^{2}}{x^{2}+4}\right)^{n} $$

Step-by-Step Solution

Answer

The series converges for -2 < \(x\) < 2 and the sum of the series, when it converges, is \(\frac{x^{2}}{4}\).

Step 1: Identify the ratio

In this problem, \(r = \frac{x^{2}}{x^{2}+4}\) . This is the ratio of the geometric series.

Step 2: Find values of \(x\) for which the series converges

The series will converge if and only if the absolute value of the ratio is less than 1. So, we need to solve the inequality |\(r\)| < 1, i.e., \(\left|\frac{x^{2}}{x^{2}+4}\right| < 1\). This simplifies to -1 < \(\frac{x^{2}}{x^{2}+4}\) < 1. Solving this inequality yields -4 < \(x^{2}\) < 4, which in turn gives -2 < \(x\) < 2. So, the values of \(x\) for which the series converges are all real numbers between -2 and 2 (excluding -2 and 2 itself).

Step 3: Function for the sum of the series

The sum of a geometric series is given by \(\frac{a}{1 - r}\), where \(a\) is the first term and \(r\) is the ratio. Here, the first term \(a\) is \(\frac{x^{2}}{x^{2}+4}\), and the ratio \(r\) is also \(\frac{x^{2}}{x^{2}+4}\). Thus, the sum \(S\) of the series can be expressed as \(S = \frac{\frac{x^{2}}{x^{2}+4}}{1 - \frac{x^{2}}{x^{2}+4}} = \frac{x^{2}}{4}\). Note that this is valid only when \(x\) is between -2 and 2 (exclusive).

Key Concepts

Ratio Test

The Ratio Test is a common method used in calculus to determine whether a series converges or diverges. Specifically, for a geometric series, the test focuses on the absolute value of the ratio between successive terms. In a geometric series, this ratio is constant, and we call it the 'common ratio' denoted as \( r \). For the series given in the exercise, the ratio is \( r = \frac{x^{2}}{x^{2}+4} \).

To apply the Ratio Test, we calculate \( |r| \) and ensure it is less than 1 for convergence. The absolute value notation \( |r| < 1 \) implies that the terms become progressively smaller, leading the series to sum to a finite number rather than diverge to infinity. Understanding the ratio test is crucial in confirming the range of convergence for a series.

Inequality Solving

Inequality solving involves determining the range of values that a variable can take on to satisfy an inequality. In this exercise, to determine the values of \( x \) for which the geometric series converges, we solve the inequality \( \left|\frac{x^{2}}{x^{2}+4}\right| < 1 \).

This inequality essentially requires finding the conditions where the expression \( \frac{x^{2}}{x^{2}+4} \) remains strictly between -1 and 1. To solve this:

• First, consider \( \frac{x^{2}}{x^{2}+4} < 1 \). Simplify this to \( -4 < x^{2} \).
• Then, consider \( \frac{x^{2}}{x^{2}+4} > -1 \), leading to \( x^{2} < 4 \).

Combining these gives \( -2 < x < 2 \). Hence, the series converges for \( x \) in this interval, excluding the endpoints. It's helpful to break down the inequality step by step to see how each part affects the solution.

Sum of Infinite Series

The sum of an infinite geometric series, where the ratio's absolute value is less than 1, can be determined using a known formula. For a geometric series with a first term \( a \) and a common ratio \( r \), the sum \( S \) of the series is given by:
\[ S = \frac{a}{1 - r} \].
In the exercise, both \( a \) and \( r \) are \( \frac{x^{2}}{x^{2}+4} \). Thus, you can calculate the sum of the series by substituting into the formula:

• First term \( a = \frac{x^{2}}{x^{2}+4} \).
• The ratio \( r = \frac{x^{2}}{x^{2}+4} \).
• The sum is therefore \( S = \frac{\frac{x^{2}}{x^{2}+4}}{1 - \frac{x^{2}}{x^{2}+4}} = \frac{x^{2}}{4} \).

This expression provides the sum of the series for \( x \) within the range \( -2 < x < 2 \). Remember, the formula is only valid when the series converges, designed for infinite sums that approach a finite limit.