Question

Use the power series $$\frac{1}{1+x}=\sum_{n=0}^{\infty}(-1)^{n} x^{n}$$ to determine a power series, centered at 0 , for the function. Identify the interval of convergence. $$ h(x)=\frac{-2}{x^{2}-1}=\frac{1}{1+x}+\frac{1}{1-x} $$

Short answer

The power series of the given function is \(h(x) = \sum_{n=0}^{\infty}(-1)^n(1 - x^{2n})\) and its interval of convergence is \(-1 < x < 1\).

Step-by-step solution

Decompose the function

Firstly, decompose the function \(h(x)\) as \(\frac{-2}{x^{2}-1}=\frac{1}{1+x}+\frac{1}{1-x}\). The right hand side of this equation is the sum of two terms which are valid for both \(\frac{1}{1+x}\) and \(\frac{1}{1-x}\).

Write in the form of power series

Now replace each term on the right-hand side using the provided power series representation to get \(h(x) = \sum_{n=0}^{\infty}(-1)^n x^n + \sum_{n=0}^{\infty}(-1)^n(-x)^n = \sum_{n=0}^{\infty}(-1)^n(1 - x^{2n}).\)

Compute the Interval of Convergence

The series converges for \(-1 < x < 1\). This is easily seen from the form of the power series \(\frac{1}{1+x}\) and \(\frac{1}{1-x}\), which are geometric with a common ratio \(x\) or \(-x\) respectively and thus converge for \(|x|<1\) or \(|-x|<1\). Combining these yields the interval of convergence as \(-1 < x < 1\).

Key concepts

Interval of Convergence

Understanding the interval of convergence is vital for working with power series. In the context of power series, this interval is the set of all real numbers for which the series converges to a real number. The power series \(\sum_{n=0}^{\infty}a_n (x-c)^n\), centered at a point \(c\), will converge only for certain values of \(x\).

For the exercise at hand, the convergence of series for \(\frac{1}{1+x}\) and \(\frac{1}{1-x}\) dictates the interval of convergence for the function \(h(x)\). Since both are geometric series, they converge when the absolute value of the common ratio is less than 1, that is, \(|x|<1\) or \(|-x|<1\) respectively. When combined, the interval of convergence for \(h(x)\) is determined to be from \(x=-1\) to \(x=1\), exclusive of the endpoints, which means \(x\) must be strictly between -1 and 1 for the series representation of \(h(x)\) to be valid.

Identifying this interval is crucial when applying the power series to practical problems, as it ensures the results obtained are reliable within the specified range.

Geometric Series

The geometric series is a cornerstone of understanding power series and their convergence. It is a series of the form \(\sum_{n=0}^{\infty}ar^n\), where \(a\) is the first term and \(r\) is the common ratio. A notable characteristic of a geometric series is that it has a simple criterion for convergence: the series converges if and only if \(|r|<1\).

In our exercise, the function \(\frac{1}{1+x}\) can be expressed as a geometric series with \(r = -x\), and similarly, \(\frac{1}{1-x}\) has \(r = x\). The critical point here is that the geometric series helps us determine where the power series for \(\frac{-2}{x^2-1}\) converges by analyzing the individual series for each part of the function. This can be particularly useful in complex analysis and differential equations where the geometric series often serves as a building block for more complicated series representations.

Series Representation of Functions

The series representation of functions is a method by which complex functions can be expressed as the sum of an infinite series of terms. This representation is invaluable in mathematical fields such as calculus and differential equations because it allows for the simplification of functions into a form that can be easily manipulated and analyzed.

For the given function \(h(x)\), the series representation is achieved by decomposing \(\frac{-2}{x^2-1}\) into \(\frac{1}{1+x}\) and \(\frac{1}{1-x}\). Both of these constituent functions can then be expressed as a power series, thanks to their geometric nature. When each term is represented as a power series and then combined, we achieve a full series representation for \(h(x)\). Such a series not only offers insight into the behavior of \(h(x)\) near the center of the series but also enables the evaluation of integrals, derivatives, and approximations of \(h(x)\) that might be otherwise intractable.