Question

Suppose that \(\sum_{n=0}^{\infty} a_{n} x^{n}\) has an interval of convergence of \((-1,1)\). Find the interval of convergence of \(\sum_{n=0}^{\infty} a_{n}\left(\frac{x}{2}\right)^{n}\).

Short answer

The interval of convergence is \((-2, 2)\).

Step-by-step solution

Understanding the series transformation

The original power series is given by \( \sum_{n=0}^{\infty} a_{n} x^{n} \) with an interval of convergence \((-1,1)\). The new series is \( \sum_{n=0}^{\infty} a_{n}\left(\frac{x}{2}\right)^{n} \). We need to find the interval of convergence for this new series.

Substituting into the new series

Substitute \( y = \frac{x}{2} \) into the original series. This transforms \( \sum_{n=0}^{\infty} a_{n} y^{n} \), which has the same form as the original series but with an input \( y \) instead of \( x \).

Finding the interval of convergence for the substitute

The interval of convergence for the series \( \sum a_n x^n \) is \( -1 < x < 1 \). When substituting \( y = \frac{x}{2} \), we need to find the interval for \( y \), so it becomes \(-1 < \frac{x}{2} < 1\).

Solving for x

Solve the inequality \(-1 < \frac{x}{2} < 1\) to find the range of \( x \). Multiply the entire inequality by 2, yielding \(-2 < x < 2\).

Determining the interval of convergence

The inequality solved in the previous step tells us that the interval of convergence for the series \( \sum_{n=0}^{\infty} a_{n}\left(\frac{x}{2}\right)^{n} \) is \(-2 < x < 2\). This is the result for convergence of the new series.

Key concepts

Power Series

A power series is a way of expressing a function as an infinite sum of terms. Each term in the series is based on a constant coefficient and is multiplied by a power of the variable. The general form of a power series can be expressed as:

• \( \sum_{n=0}^{\infty} a_n x^n \), where \( a_n \) are constant coefficients, and \( x \) is the variable.

Power series are quite handy because they allow us to represent functions that are otherwise difficult to analyze. These series have an interval of convergence, which is the set of all \( x \)-values for which this infinite sum results in a finite number. Understanding the interval of convergence for a power series is crucial, as it tells us where the series accurately represents the function.Remember, just because a series converges at some \( x \)-value doesn't mean it converges at all nearby \( x \)-values. It is always important to find the interval of convergence when working with power series.

Series Transformation

Series transformation involves altering a given power series into a new series. This could involve changing the variable or applying some operation to the coefficients or terms. In our example, we deal with transforming the original series:

• From \( \sum_{n=0}^{\infty} a_{n} x^{n} \)
• To \( \sum_{n=0}^{\infty} a_{n}\left(\frac{x}{2}\right)^{n} \)

Transforming a series often affects its interval of convergence. Therefore, it is necessary to carefully assess how such a transformation impacts convergence. For instance, substituting \( \frac{x}{2} \) changes only the variable part, and that's what's being solved for. Understanding the effects of transformation on the interval enables us to manage and predict the behavior of the series effectively as it changes.

Inequality Solving

Inequality solving is a technique used to determine the values or range of values that a certain expression can take. In context with power series, inequalities help us explore the interval of convergence.In our transformed series example, we face the inequality:

• \(-1 < \frac{x}{2} < 1\)

To solve this inequality, we perform arithmetic operations to isolate \( x \). First, we multiply all parts of the inequality by 2, which helps eliminate the fraction:

• Results in \(-2 < x < 2\)

By solving this inequality, we established that the series converges for \( x \) within the range of \(-2 < x < 2\). Solving inequalities is indispensable in mathematical problem-solving as it facilitates understanding of the limitations and behavior of functions and series within specified conditions.

Substitution Method

The substitution method is an approach used to simplify and solve complex mathematical expressions or equations. This technique entails replacing a variable with another expression or variable to make calculations more manageable.Consider our example, where substitution was crucial. The original power series was in terms of \( x \), and by substituting \( y = \frac{x}{2} \), the series is transformed into a familiar form:

• \( \sum_{n=0}^{\infty} a_{n} y^{n} \)

Using substitution can simplify the problem, making it possible to apply known convergence criteria or other analytical methods. It also aligns the new series with known conditions such as convergence intervals. Substitution makes it easier to carry out further calculations or transformations and is a powerful tool in mathematics to aid in clarity and solve problems efficiently.