Question

Combining Power Series Suppose that \(\sum_{n=0}^{\infty} a_{n} x^{n}\) is a power series whose interval of convergence is \((-1,1)\), and suppose that \(\sum_{n=0}^{\infty} b_{n} x^{n}\) a power series whose interval of convergence is \((-2,2)\). a. Find the interval of convergence of the series \(\sum_{n=0}^{\infty}\left(a_{n} x^{n}+b_{n} x^{n}\right)\). b. Find the interval of convergence of the series \(\sum_{n=0}^{\infty} a_{n} 3^{n} x^{n}\).

Short answer

a. Interval: (-1, 1). b. Interval: \((-\frac{1}{3}, \frac{1}{3})\).

Step-by-step solution

Understanding the Problem

We are given two power series with different intervals of convergence, and we need to find the intervals of convergence for their combined series in two different forms. We will handle each question separately.

Determine Combined Series Interval (a)

Both series, \(\sum_{n=0}^{\infty} a_{n} x^{n}\) and \(\sum_{n=0}^{\infty} b_{n} x^{n}\), will converge absolutely within their overlapping interval of convergence. Since the convergence for each individual series is independent, the shortest interval \((-1, 1)\) must be considered for convergence. Therefore, the interval of convergence for \(\sum_{n=0}^{\infty} (a_{n} x^{n} + b_{n} x^{n})\) is \((-1, 1)\).

Analyze Interval with Scaling (b)

For the series \(\sum_{n=0}^{\infty} a_{n} 3^n x^{n}\), we need to modify the interval of convergence based on the substitution of \(3x\) into the original power series. The original series \(\sum_{n=0}^{\infty} a_{n} x^{n}\) converges for \(-1 < x < 1\). If we replace \(x\) by \(3x\), the interval changes to \(-1 < 3x < 1\). Solving for \(x\), we divide through by 3: \(-\frac{1}{3} < x < \frac{1}{3}\). So, the interval of convergence for \(\sum_{n=0}^{\infty} a_{n} 3^n x^{n}\) is \((-\frac{1}{3}, \frac{1}{3})\).

Key concepts

Power Series

A power series is a mathematical series of the form \( \sum_{n=0}^{\infty} a_n x^n \), where each term involves a power of a variable, often denoted as \( x \). Power series are vital because they allow functions to be expressed as infinite sums, making complicated functions easier to analyze.

• The \'\( a_n \)'s are the coefficients, which can be any real or complex numbers, providing the series with its unique signature.
• \( x \) is the variable, which is raised to the power \( n \).
• The interval of convergence is the set of values of \( x \) for which the series converges to a finite number.

For a given power series, the interval of convergence can be found using the ratio test, which gives insight into the nature of convergence and is calculated by examining where the terms become negligible. Understanding intervals of convergence is crucial because it tells us the range of \( x \) for which the series representation of the function is valid.

Absolute Convergence

Absolute convergence refers to a series \( \sum_{n=0}^{\infty} a_n \) that converges even when all of its terms are replaced by their absolute values, \( \sum_{n=0}^{\infty} |a_n| \). If a series converges absolutely, it implies a robust form of convergence.

• Absolute convergence ensures the series retains convergence regardless of term rearrangement.
• An absolutely convergent series also converges in the standard sense.

When dealing with power series, absolute convergence within an interval implies that rearranging or combining power series terms will not affect convergence. For example, when combining two power series \( \sum_{n=0}^{\infty} a_n x^n \) and \( \sum_{n=0}^{\infty} b_n x^n \), we must consider the interval in which both series are absolutely convergent to ensure convergence of the new series \( \sum_{n=0}^{\infty} (a_n + b_n) x^n \). Hence, it becomes essential to find the smaller interval of convergence common to both for certainty of convergence.

Substitution in Series

Substitution in series can modify the range of the interval of convergence by altering the variable term, effectively transforming the series. For instance, consider a power series \( \sum_{n=0}^{\infty} a_n x^n \) with an interval of convergence \((-1,1)\).

• When substituting \( x \) with \( c \cdot x \), the new interval is determined by solving \(-1 < c \cdot x < 1\), giving \( -\frac{1}{c} < x < \frac{1}{c} \).
• This method can narrow or shift the interval, reflecting the new domain then covered by the convergent series.

A practical application is seen when scaling a series \( \sum_{n=0}^{\infty} a_n 3^n x^n \), acting like a substitution with \( 3x \), shifting the interval of convergence. It's essential to solve the inequality for \( x \) to find the appropriate interval accurately after substitution, ensuring correctness and understanding of the result's new form.