Problem 9 - Essential Calculus

Question

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

Step-by-Step Solution

Answer

The power series for the function \(f(x) = \frac{3}{x+2}\) centered at 0 is given by \(3 \sum_{n=0}^{\infty}(-1/2)^n x^n\). The interval of convergence for this power series is \((-2, 2)\).

Step 1: Rewrite the function

Start by rewriting the function in a form where the geometric series formula can be applied. In this case, the geometric series is \(\frac{1}{1-r}\), where \(r\) is the ratio. The given function is \(f(x)=\frac{3}{x+2}\), thus we want to bring it to a form \(\frac{3}{1+(-x/2)}\).

Step 2: Apply the geometric series formula

Now, apply the power series expansion of \(\frac{1}{1-y}\), which is \(\sum_{n=0}^{\infty}y^n\), where our \(y = -x/2\). So we have \(\frac{3}{1+(-x/2)} = 3 \sum_{n=0}^{\infty}(-x/2)^n\).

Step 3: Simplify the series

The series right now has a fraction inside. We need to simplify it to get a nicer form. Doing that, we get \(3 \sum_{n=0}^{\infty}(-1/2)^n x^n\).

Step 4: Determine the interval of convergence

Now, the ratio test can be applied to determine the interval of convergence. In general, \(\left| \frac{a_{n+1}}{a_n} \right| < 1\) gives the radius of convergence \(R\). Thus, the interval of convergence will be \((-2R, 2R)\). In this case, our \(a_n = (-1/2)^n\) and our radius of convergence \(R = 1\), so the interval of convergence is \((-2, 2)\).

Key Concepts

Geometric Series

A geometric series is a sum of terms that each differ from the previous one by a common ratio. For example, in the series \(1, r, r^2, r^3, \dots\), each term after the first is obtained by multiplying the previous one by the ratio \(r\). The sum of an infinite geometric series can be found using the formula \( \sum_{n=0}^{\infty} r^n = \frac{1}{1 - r} \), provided that the absolute value of the ratio is less than one, \( |r| < 1 \). This condition ensures that the terms get smaller and the series converges to a finite value.
When finding a power series representation of a function, like in the exercise \(f(x)=\frac{3}{x+2}\), we often look to express the function in a form that allows us to use this geometric series formula. By rewriting the function appropriately and identifying the ratio, we can expand the function into an infinite series.

Interval of Convergence

The interval of convergence is the set of all real numbers for which a power series converges to a finite value. It's critical to understand that power series do not necessarily converge for all values of \(x\).
For the power series \( \sum_{n=0}^{\infty} a_n (x - c)^n \), where \(a_n\) represents the coefficients and \(c\) is the center of the series, the interval of convergence is centered around that \(c\). It can be a finite interval, a single point, or the entire set of real numbers. In the provided exercise, once the series is determined to be geometric, the interval of convergence is found using the ratio test, resulting in a finite interval from \( (-2, 2) \) for the given function.

Ratio Test

The ratio test is a tool to determine the convergence of infinite series. Specifically for power series, it helps in finding the radius of convergence, which directly relates to the interval of convergence. By examining the limit of the absolute value of the ratio of consecutive terms, \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \), we can conclude about the series' behavior:

• If the limit is less than one, the series converges.
• If the limit is greater than one or diverges, the series does not converge.
• If the limit equals one, the test is inconclusive and other methods must be used.

Applying this to our exercise where \(a_n = (-1/2)^n\), the ratio test confirms that the series converges within the interval \( (-2, 2) \).

Series Expansion

Series expansion is the process of expressing a function as the sum of an infinite series. In calculus, it is commonly done through power series, where a function is written in the form of \( \sum_{n=0}^{\infty} a_n (x - c)^n \). The coefficients \(a_n\) and the terms \( (x - c)^n \) are derived from the function itself. For functions that can be described by a geometric series, the series expansion process involves rewriting the function into a form that reveals the recurrent multiplication factor—such as \(3 \sum_{n=0}^{\infty}(-1/2)^n x^n\) from our original exercise. Once expanded, these series can be used to approximate functions, solve differential equations, or model complex phenomena within an interval where the series converges.