Question

Find a power series for the function, centered at \(c,\) and determine the interval of convergence. $$ g(x)=\frac{3 x}{x^{2}+x-2}, \quad c=0 $$

Short answer

The power series representation for \( g(x)=\frac{3 x}{x^{2}+x-2} \) centered at 0 is \(g(x) = 1 + x + x^{2}+ x^{3} + ... + 2 - x + x^{2}/2 - x^{3}/8 + ...\). The interval of convergence is \(-2 < x < 1\).

Step-by-step solution

Rewrite the function in a form that can be recognized as a geometric series

The first step in this process is to rewrite the function in a form that can be recognized as a geometric series. We want to get it into the form \( \frac{A}{1 - x}\). To do that, first we factorize the denominator of \(g(x)\), which results in \(x^{2}+x-2 = (x - 1)(x + 2)\) and express \(g(x)\) as partial fractions. We get \[g(x) = \frac{1}{x - 1} + \frac{2}{x + 2}.\]

Expand each term into a power series

Now with the form \(\frac{A}{1 - x}\) for each term, each term of \(g(x)\) can be expanded into a power series centered at \(c = 0\) using the formula for the sum of a geometric series \(\frac{a}{1 - r} = a + ar + ar^{2} + ar^{3} + ...\). For the first term, \(x\) is replaced with \(-x\) to fit the form. So we have:\-For the first term: \[\frac{1}{1 - x} = 1 + x + x^{2} + x^{3}+ ...\]\-For the second term: \[\frac{2}{1 - (-x/2)} = 2 - x + x^{2}/2 - x^{3}/8 + ...\]So the power series for \(g(x)\) centered at \(c = 0\) is \(g(x) = 1 + x + x^{2} + x^{3}+ ... + 2 - x + x^{2}/2 - x^{3}/8 + ...\).

Determine the interval of convergence

The interval of convergence is determined by the absolute value of \(x\) being less than 1 for the geometric series to converge, and it needs to satisfy for both terms. Here, \(x < 1\) and \(-x/2 < 1\). Solving both gives the interval \(-2 < x < 1\). This is the open interval of convergence for the power series.

Key concepts

Geometric Series

Understanding the concept of a geometric series is critical when dealing with power series representations of functions. A geometric series is a sum of the form \( a + ar + ar^2 + ar^3 + \ldots \), where \(|r| < 1\). The series converges to the sum \( S = \frac{a}{1 - r} \) if the absolute value of the common ratio \(|r|\) is less than one.

Geometric series are particularly useful for rewriting functions into power series because they allow us to represent a function as an infinite sum of terms with increasing powers. This capacity becomes a powerful tool in calculus for approximations and solving differential equations.

When working with power series and geometric series, you may notice that the convergence hinges on the variable \(|x|\) being within a certain range. This is why the concept of the interval of convergence, which is discussed in a separate section, should always be determined.

Partial Fraction Decomposition

Partial fraction decomposition is the process of breaking down a rational function into a sum of simpler fractions, which can then be individually expanded into power series. This technique is particularly useful when dealing with complex rational functions like \frac{3x}{x^2+x-2}\, where direct expansion into a power series is not straightforward.

In our step by step solution, the function \( g(x) \) was decomposed into \frac{1}{x - 1}\ and \frac{2}{x + 2}\, allowing us to deal with each fraction separately. This decomposition is based on factoring the denominator, which can often lead to a more manageable analysis of the function's behavior across the domain, and sets the stage for finding a power series centered at a specific point, such as \c = 0\ in this case.

Mastering partial fraction decomposition is vital, as it not only aids in the expansion into power series but also simplifies integrals, differential equations, and other complex calculus operations.

Interval of Convergence

The interval of convergence is an essential concept when discussing power series. It refers to the set of all \x\-values for which the series converges, which means the series will add up to a finite number.

To find this interval, we must identify the range of \x\-values that satisfy the condition of convergence. In our solution, the condition for convergence of each series was that the absolute value of the respective terms' common ratio must be less than 1. This condition was translated into inequalities: \(-2 < x < 1\). These inequalities define the open interval of convergence, meaning that the series converges to a function within this range. It's crucial to test the endpoints separately because the behavior of the series can differ at these points, sometimes leading to convergence at one endpoint, both, or neither.

Understanding the significance of the interval of convergence aids in comprehending the limits within which a power series representation of a function is valid. This ensures that the function described by the series corresponds to the original function within this specific range.