Problem 34 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Use any method to find the Maclaurin series for \(f(x) .\) (Strive for efficiency.) Determine the radius of convergence. $$f(x)=\frac{1}{2 x+3}$$

Step-by-Step Solution

Answer

The Maclaurin series for the function \(f(x) = \frac{1}{2x + 3}\) is \(\frac{1}{3} + \frac{2x}{3} + \frac{4x^2}{9} + \frac{8x^3}{27} + \cdots\). The radius of convergence is \(R = 1.5\).

Step 1: Apply the Geometric Series

Rewrite \(f(x)\) in the form of \(a /(1-r)\) where \(a = 1/3\) and \(r = 2x/3\). This allows us to use the geometric series formula, which states that \(1 /(1-r) = 1 + r + r^2 + r^3 + \cdots\)

Step 2: Form the Maclaurin Series

Substitute \(a\) and \(r\) into the geometric series to obtain the Maclaurin series: \(f(x) = 1/3 + (2x/3) + (4x^2/9) + (8x^3/27) + \cdots = \sum_{n=0}^{\infty} (2/3)^n x^n\)

Step 3: Determine the Radius of Convergence

For a geometric series, the absolute value of the common ratio is less than 1 for the series to converge. So \(|2x/3| < 1\), which implies the radius of convergence \(R\) is 1.5.

Key Concepts

geometric series

A geometric series is a series of terms in which each term is a constant multiple of the previous one. In simpler terms, it is a sequence where you multiply by the same number each time to get from one term to the next.

In mathematical notation, a geometric series can be written as:

• First term: \(a\)
• Common ratio: \(r\)
• Series: \(a, ar, ar^2, ar^3, \ldots\)

This series can continue indefinitely, and the sum can be described with the formula:\[S = \frac{a}{1 - r},\]as long as the absolute value of \(r\) is less than 1.

This concept is fundamental when dealing with functions you want to express as a series, particularly over a small interval around a point, like in the Maclaurin series. It simplifies the manipulation of the function into an infinite sum of terms, each of which is a power of \(x\). This is incredibly useful in calculus for functions that can be fit into this pattern.

radius of convergence

The radius of convergence is a key concept when discussing the convergence of power series. It is essentially the distance from the center of the series within which the series will converge.

For a series to converge, the terms must get smaller and approach zero as the series goes on. For a geometric series that follows the form \(\sum_{n=0}^{\infty} ar^n\), it converges when the absolute value of \(r\) is less than 1: \[ |r| < 1.\]Here, the term \( |2x/3| = |r| < 1 \) leads to the radius of convergence \(R\). Solving the inequality, we find that the radius of convergence for the given series is 1.5. This means the series expressed in the Maclaurin series format converges when \(|x| < 1.5\).

Practically, this gives us insight into the interval in which we expect our entire series to provide accurate approximations of our function \(f(x)\). Understanding and determining the radius of convergence is crucial for accurate mathematical modeling.

convergent series

In mathematics, a convergent series is a series whose terms approach a specific value as you sum more and more of them. This means that even though you may be adding an infinite number of terms, the overall sum approaches a finite limit.

A simple criterion for a series \(\sum a_n\) to be convergent is that the terms \(a_n\) must approach zero as \(n\) approaches infinity. Convergent series can be seen as the backbone of functions expressed as series, allowing them to be approximated by a finite number of terms in practical applications.

For the Maclaurin series example discussed, convergence depends on the absolute value of the ratio \(r\), where for convergence, \(|r| < 1\). Continuing the series results only if each term becomes smaller, leading to the infinite sum converging to a finite, specific value.

The utility of understanding whether a series converges lies in its application to solving equations, running computations, and modeling various real-world scenarios where direct calculation or formula manipulation could be too complex or impossible.