How to Find Interval of Convergence for Power Series
Power series are an essential tool in mathematics, particularly in the study of functions and their properties. One of the fundamental questions that arise when dealing with power series is how to determine their interval of convergence. This article aims to provide a comprehensive guide on finding the interval of convergence for power series, covering various methods and techniques.
Understanding the Basics
Before diving into the methods for finding the interval of convergence, it is crucial to understand the basic concept. A power series is an infinite series of the form:
$$\sum_{n=0}^{\infty} a_n (x-c)^n$$
where \(a_n\) are constants, \(x\) is the variable, and \(c\) is the center of the series. The interval of convergence is the set of all values of \(x\) for which the series converges.
Ratio Test
The ratio test is a popular method for determining the interval of convergence. It involves calculating the limit of the absolute value of the ratio of consecutive terms in the series:
$$\lim_{n \to \infty} \left|\frac{a_{n+1}(x-c)^{n+1}}{a_n(x-c)^n}\right|$$
If the limit is less than 1, the series converges; if the limit is greater than 1, the series diverges; and if the limit is equal to 1, the test is inconclusive.
Example
Consider the power series:
$$\sum_{n=0}^{\infty} \frac{(x-2)^n}{3^n}$$
To find the interval of convergence, we apply the ratio test:
$$\lim_{n \to \infty} \left|\frac{\frac{(x-2)^{n+1}}{3^{n+1}}}{\frac{(x-2)^n}{3^n}}\right| = \lim_{n \to \infty} \left|\frac{x-2}{3}\right|$$
The series converges when the limit is less than 1, which gives us the inequality:
$$\left|\frac{x-2}{3}\right| < 1$$ Solving this inequality, we find that the interval of convergence is: $$(1, 5)$$
Cauchy-Hadamard Formula
The Cauchy-Hadamard formula is another method for finding the interval of convergence. It involves calculating the radius of convergence, which is the distance from the center of the series to the nearest boundary of the interval of convergence. The formula is given by:
$$R = \frac{1}{\limsup_{n \to \infty} \sqrt[n]{|a_n|}}$$
If \(R\) is finite, the interval of convergence is \((c-R, c+R)\). If \(R\) is infinite, the series converges for all \(x\), and if \(R\) is 0, the series converges only at \(x=c\).
Example
Using the Cauchy-Hadamard formula for the same power series as before:
$$R = \frac{1}{\limsup_{n \to \infty} \sqrt[n]{\left|\frac{1}{3^n}\right|}} = \frac{1}{\frac{1}{3}} = 3$$
Therefore, the interval of convergence is:
$$(2-3, 2+3) = (-1, 5)$$
Conclusion
Finding the interval of convergence for power series is a crucial step in understanding the behavior of functions represented by these series. By applying methods such as the ratio test and the Cauchy-Hadamard formula, we can determine the values of \(x\) for which a power series converges. This knowledge is invaluable in various fields of mathematics, including analysis, physics, and engineering.
