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Chapter 9: Problem 68

True or false. Give an explanation for your answer. \(-3 < x < 2\) could be the interval of convergence of \(\sum C_{n} x^{n}\)

Short Answer

Expert verified
True; the interval \(-3 < x < 2\) can be an interval of convergence for a power series.

Step by step solution

01

Understand Series Convergence

A power series \( \sum C_{n} x^{n} \) is given. Its interval of convergence refers to the range of \( x \) values for which the series converges.
02

Interval Characteristics

The interval of convergence can be open, closed, or half-open (either \((a,b), [a,b), (a,b], or [a,b])\). It includes all \( x \) values where the series converges.
03

Analyze Given Interval

The given interval is \(-3 < x < 2\), which is an open interval from \(-3\) to \(2\). This indicates the series converges within this range but excludes the endpoints \(-3\) and \(2\).
04

Determine Possibility Based on Theory

An interval \((-3, 2)\) could be a legitimate interval of convergence as power series can have open intervals of convergence. Typically, convergence at endpoints may vary; thus the series can converge within such an open interval.
05

Conclusion

The statement is true. Given that \(-3 < x < 2\) is an open interval, it is possible for a power series鈥� interval of convergence to be \((-3, 2)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Series
A power series is an infinite series of the form \[ \sum_{n=0}^{\infty} C_{n} x^{n} \]where \(C_{n}\) are coefficients and \(x\) is the variable. Each term in the series has \(x\) raised to an increasing power, starting from zero. The power series is a powerful tool in mathematical analysis and appears frequently in calculus and differential equations.
  • The coefficients \(C_{n}\) can be any real numbers.
  • \(x\) is a variable that can take values in a particular range where the series converges.
  • The series represents a function within its interval of convergence.
Understanding where this series converges or diverges is key; this is determined by the interval of convergence. Working with power series allows mathematicians to approximate more complex functions or solve differential equations effectively.
Convergence
Convergence refers to whether a series like the power series settles into a particular value as more terms are added. Not all series will converge. A series that converges does so when its sum approaches a finite limit when infinitely many terms are added. For a power series \(\sum C_{n} x^{n}\), convergence can be found using techniques like the Ratio Test or the Root Test.
  • For the Ratio Test, examine \(\lim_{n \to \infty} \left| \frac{C_{n+1} x^{n+1}}{C_{n} x^{n}} \right| \).
  • If this limit is less than 1, the series converges.
  • The Root Test involves \(\lim_{n \to \infty} \sqrt[n]{|C_{n} x^{n}|}\).
The range of \(x\) values for which the series converges comprises the interval of convergence. This concept is crucial for determining how a power series behaves and whether it can be used to represent a function in a specific range.
Open Interval
An open interval is a set of numbers that includes all real numbers between two bounds but not the bounds themselves. Represented as \((a, b)\), this interval contains all \(x\) satisfying \(a < x < b\). In the context of power series, the interval of convergence can often be an open interval.
  • Open intervals are useful for understanding where a series converges while excluding endpoints.
  • This kind of interval implies different behavior if analyzed towards its endpoints.
  • Convergence at the endpoints needs to be tested separately, as it does not automatically result from the series converging within the open interval.
The example provided, \(-3 < x < 2\), represents an open interval. It's an excellent illustration of where a power series might converge: within the interval, but not necessarily at its endpoints \(-3\) and \(2\). Understanding the nature of open intervals helps in properly analyzing the behavior of series like \(\sum C_{n} x^{n}\).

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Most popular questions from this chapter

True or false. Give an explanation for your answer. If the power series \(\sum C_{n} x^{n}\) converges for \(x=2,\) then it converges for \(x=1\)

True or false. Give an explanation for your answer. \(\sum C_{n}(x-1)^{n}\) and \(\sum C_{n} x^{n}\) have the same radius of convergence.

Determine whether the series converges. $$\sum_{n=1}^{\infty} \frac{\cos (n \pi)}{n}$$

Determine whether the series converges. $$\sum_{n=2}^{\infty} \frac{3}{\ln n^{2}}$$

Decide if the statements are true or false. Give an explanation for your answer. If \(b_{n} \leq a_{n} \leq 0\) for all \(n\) and \(\sum b_{n}\) converges, then \(\sum a_{n}\) converges.
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