Problem 40 - Calculus: Early Transcendentals

Question

Suppose that the power series \( \sum c_n (x - a)^n \) satisfies \( c_n \not= 0 \) for all \( n. \) Show that if \( \lim_{n \to \infty} \mid c_n/c_{n + 1} \mid \) exists, then it is equal to the radius of convergence of the power series.

Step-by-Step Solution

Answer

The radius of convergence is equal to \( \lim_{n \to \infty} |c_n/c_{n+1}| \), denoted by \( R = L \).

Step 1: Define the Ratio Test

The ratio test states that for a power series \( \sum c_n (x-a)^n \), the radius of convergence \( R \) can be determined by evaluating the limit \( L = \lim_{n \to \infty} \left| \frac{c_n}{c_{n + 1}} \right| \). If this limit exists, then \( R = L \).

Step 2: Apply the Ratio Test Limit

Given that \( \lim_{n \to \infty} \left| \frac{c_n}{c_{n + 1}} \right| \) exists, we denote this limit by \( L \). According to the ratio test, the radius of convergence \( R \) is determined by \( R = L \). This relationship directly follows from the definition when the limit exists.

Step 3: Conclude Relationship of Limit and Radius

Since it is given that \( c_n eq 0 \) for all \( n \) and \( \lim_{n \to \infty} \left| \frac{c_n}{c_{n + 1}} \right| = L \) exists, by the ratio test, the radius of convergence \( R \) for the series \( \sum c_n (x-a)^n \) is \( R = L \), which is the established formula. There is a direct correspondence between this limit and the radius based on the infinite series convergence criteria.

Key Concepts

Power Series

A power series is an infinite series of the form \( \sum c_n (x-a)^n \), where \( c_n \) are the coefficients, \( x \) is the variable, and \( a \) is the center of the series. These series play a crucial role in calculus and analysis because they help represent complex functions in simpler forms by breaking them down into an infinite sum of polynomials.
Power series are especially useful because of their flexibility. They can converge (or "settle down to a value") only for certain values of \( x \). This set of \( x \) values is determined by the radius of convergence, which we'll explore further. In more simple terms, a power series can be thought of as a mathematical tool, sorting numbers to see which ones work to create a continuous function when added up.
A particularly interesting aspect of power series is that they can approximate functions extremely well within their interval of convergence, providing an accessible way to tackle complex problems in mathematics and physics.

Ratio Test

The Ratio Test is a method used to determine the convergence or divergence of an infinite series, especially useful for power series. For the series \( \sum c_n (x-a)^n \), the Ratio Test evaluates the terms \( c_n \) by looking at the ratio of successive terms: \( \lim_{n \to \infty} \left| \frac{c_n}{c_{n + 1}} \right| \).
If this limit exists and is less than, greater than, or equal to 1, it helps decide on the series' behavior.

• If the limit is less than 1, the series converges absolutely; this means that no matter how large \( n \) gets, the terms will decrease in size.
• If the limit is greater than 1, the series diverges; this means the terms do not decrease sufficiently for the series to add up to a finite number.
• If the limit equals 1, the test is inconclusive; other methods are needed to determine convergence.

The Ratio Test is especially effective for determining the radius of convergence \( R \) for power series. If \( \lim_{n \to \infty} \left| \frac{c_n}{c_{n + 1}} \right| = L \), then the radius of convergence is \( R = L \). This knowledge enables us to predict within what interval our power series can be expressed as a finite function.

Infinite Series Convergence

Infinite series convergence is the concept where an infinite sequence of numbers has a well-defined sum. Unlike finite sums which are straightforward, whether an infinite series converges can sometimes be more challenging to determine.
An infinite series such as \( \sum c_n \) converges when the sum of its infinite terms approaches a finite limit as more and more terms are added. This only happens when the series terms get smaller and smaller, often approaching zero.

• For a power series like \( \sum c_n (x-a)^n \), convergence depends on the chosen \( x \) and the radius of convergence \( R \).
• If \( |x-a| < R \), the series converges.
• If \( |x-a| > R \), the series diverges.
• For \( |x-a| = R \), convergence or divergence must be determined individually, as it can vary.

Understanding convergence is crucial because it dictates where a series accurately represents a function. Familiarity with convergence and divergence ensures that one can utilize power series effectively in numerous mathematical applications.