Question

Find a power series for the function, centered at \(c,\) and determine the interval of convergence. $$ f(x)=\frac{3}{2 x-1}, \quad c=2 $$

Short answer

The power series representation of \(f(x) = \frac{3}{2x - 1}\) centered at \(c = 2\) is \(f(x) = \sum_{n=0}^{\infty} 3*(\frac{1}{2})^n * (x-2)^n\) and the interval of convergence is [0, 4].

Step-by-step solution

Apply the formula for a geometric series

First, observe that \(f(x)\) can be written as a geometric series. We can rewrite \(f(x)\) as \(f(x) = 3 * \frac{1}{2x-2}\), which is in the form of a geometric series, where \(r = \frac{(x-2)}{2}\). So, \(f(x)\) can be written as \(f(x) = \sum_{n=0}^{\infty} 3* (\frac{1}{2})^n * (x-2)^n\). Therefore, the power series representation of \(f(x)\) is \(f(x) = \sum_{n=0}^{\infty} 3* (\frac{1}{2})^n * (x-2)^n\).

Find the Interval of Convergence

To find the interval of convergence, we apply the Ratio Test which is finding the limit: \(\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}|\) and for convergence, it should be less than 1. Setting \((x-2\)/2\) equal to 1 and -1 gives x = 4 and x = 0. This method helps to find the radii of convergence. We can recognize that the radius of convergence is '2' and about \(x = 2\), the interval of convergence is [0, 4]. After determining the endpoints of the interval, a test must be made to determine whether they're included in the interval of convergence. Here, the end points are included. Hence the Interval of Convergence is [0, 4].

Key concepts

Geometric Series

Understanding the concept of a geometric series is crucial when dealing with power series representations of functions. A geometric series is a series of terms where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (denoted as r). Mathematically, it's expressed as:

\[ a + ar + ar^2 + ar^3 + \.\.\. + ar^n \]
Where a is the first term of the series. Such a series converges if the absolute value of the common ratio |r| < 1. If <|r|>1>, the series diverges, growing without bound. In the context of power series, as shown in the exercise, the function f(x) can be expressed as an infinite geometric series centered at a value c, providing a powerful tool to analyze complex functions in a simpler form.

Interval of Convergence

The interval of convergence is fundamentally important for determining the set of x-values for which a power series converges to the function it represents. This interval indicates the range within which the power series converges, and outside of which, it diverges. In the given exercise, the interval of convergence for the power series representation of f(x) was determined to be between 0 and 4, including these endpoints. This means that for any x in the closed interval [0, 4], the infinite series converges and f(x) can be expressed as that series. It's essential to test the endpoints separately because convergence at these points isn't guaranteed solely by the radius of convergence.

When finding the interval of convergence for any power series, one must always analyze the behavior at the endpoints through tests such as the Ratio Test or by directly plugging in the values into the power series.

Ratio Test

The Ratio Test is an invaluable tool in determining the convergence of series, especially when dealing with power series. This test involves taking the limit of the absolute value of the ratio of the (n+1)th term a_(n+1) to the nth term a_n as n approaches infinity. The key formula is:

\[ \text{Let } L = \text{lim}_{n \to \text{infinity}} | \text{frac}{a_{n+1}}{a_n} | \]
If L < 1, the series converges; if L > 1, it diverges; and if L = 1, the test is inconclusive. In our original example, by simplifying the function into a geometric series form and applying the Ratio Test, we found that the series representation converges for the absolute value of (x-2) / 2 less than 1, which corresponds to an interval centered at c = 2. This test thus quantifies the series' behavior and aids in identifying the radius and interval of convergence with clarity.

Radius of Convergence

The radius of convergence ties together the concepts of power series, geometric series, and convergence. It specifically relates to the distance from the center point of the series (in our example, c = 2) to the edge of the interval within which the series converges. For a power series, the radius of convergence R can be found by examining the limit used in the Ratio Test.

For the geometric series represented as f(x) = sum_{n=0}^infinity 3 * (1/2)^n * (x-2)^n, we determine the radius of convergence by setting the common ratio | ((x-2)/2) | < 1, which gives us the inequalities 0 < (x-2) < 4. Hence, R = 2 is the radius, and the interval of convergence is [c-R, c+R], or in our case, [0, 4], including the endpoints. This concept is vital for students to identify the scope within which the power series is an effective representation of the function.