Question

Find a power series for the function, centered at \(c,\) and determine the interval of convergence. $$ f(x)=\frac{2}{1-x^{2}}, \quad c=0 $$

Short answer

The power series representation of the function \( f(x)=\frac{2}{1-x^{2}} \) is \( f(x) = 2\sum_{n=0}^{\infty} x^{2n} \) and the interval of convergence is \(-1,x,1\).

Step-by-step solution

Identify the Known Power Series

The function is given as \( f(x)=\frac{2}{1-x^{2}} \). Rewrite this function into a form that matches the known power series for a geometric series \( \frac{1}{1-x} \). After rewriting it becomes \( f(x)=2\sum_{n=0}^{\infty} (x^2)^n = 2\sum_{n=0}^{\infty} x^{2n} \). This power series is centered at \(c=0\).

Calculate the Interval of Convergence

Now, we determine the interval of convergence by using the ratio test for which the absolute value of the ratio between the \((n+1)\)th term and the nth term of the power series is less than 1. After applying the ratio test, we find \( |x^2| < 1 \). It means that the power series converges for \( -1 < x < 1 \), therefore the interval of convergence is \(-1,1\).

Presenting the Final Answer

The function \( f(x)=\frac{2}{1-x^{2}} \) can be expressed in terms of a power series as \( f(x) = 2\sum_{n=0}^{\infty} x^{2n} \), for \( -1 < x < 1 \).

Key concepts

Interval of Convergence

The concept of an interval of convergence is crucial in understanding the behavior of power series. In the context of power series, convergence refers to the series approaching a finite value as more terms are added. However, the series will not converge for all values of the variable. The interval of convergence is the range of values for which the series converges to a finite number.

As evidenced in the exercise for the function \( f(x)=\frac{2}{1-x^{2}} \), we can express it as a power series centered at 0. To determine its interval of convergence, we must find where this power series yields valid results. The ratio test, which is further elaborated under its dedicated section, helps us to establish that \( -1 < x < 1 \) is this series’ interval of convergence. Within this interval, the function can be represented by its corresponding power series with mathematical precision.

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In terms of power series, when we speak of a geometric series, we often mean a series of the form \( \frac{1}{1-x} = \sum_{n=0}^{\text{\infty}} x^n \). This series is only convergent when the absolute value of \( x \) is less than one, showcasing a direct link to the interval of convergence.

In the given exercise, the function \( f(x) \) is manipulated to resemble a geometric series by considering \( x^2 \) as the common ratio. This enables us to extract a power series representation of the function, leading to the final answer, and also sets a good foundation for understanding the broader category of power series, to which geometric series belongs.

Ratio Test

The ratio test is a decisive tool for determining the convergence of infinite series, especially power series. The test evaluates the limit of the absolute value of the ratio of successive terms in the series. If this limit is less than 1, the series converges; if more than 1, it diverges; and if equal to 1, the test is inconclusive.

Applying the ratio test to the exercise's power series \( 2\sum_{n=0}^{\text{\infty}} x^{2n} \) leads to examining the limit of \( |\frac{x^{2(n+1)}}{x^{2n}}| \). Simplification of this ratio gives us \( |x^2| \) and, upon applying the ratio test, we are left with the inequality \( |x^2| < 1 \), which clarifies the boundary for the interval of convergence being \( -1 < x < 1 \). This test is not only essential for concluding convergence but also provides a systematic method applicable to a wide array of series beyond the scope of the exercise.