Question

If the radius of convergence of \(\sum_{n=0}^{\infty} a_{n} x^{n}\) is \(5,\) what is the radius of convergence of \(\sum_{n=0}^{\infty}(-1)^{n} a_{n} x^{n} ?\)

Short answer

The radius of convergence remains 5.

Step-by-step solution

Understanding the Problem

We are given a power series \( \sum_{n=0}^{\infty} a_{n} x^{n} \) with a radius of convergence of 5. We need to find the radius of convergence of a similar power series \( \sum_{n=0}^{\infty} (-1)^{n} a_{n} x^{n} \).

Identifying Key Principles

The radius of convergence for a power series \( \sum_{n=0}^{\infty} a_{n} x^{n} \) is determined by the growth of its coefficients \( a_n \). Multiplying the terms by \( (-1)^{n} \) doesn't affect their growth rates, just their signs.

Determine the Radius of Convergence

Since altering the sign of the terms \( a_n \) to \( (-1)^n a_n \) does not affect their growth rate, the radius of convergence remains determined by the same criteria. Thus, the radius of convergence of both series is the same.

Conclusion

The radius of convergence of \( \sum_{n=0}^{\infty}(-1)^{n} a_{n} x^{n} \) is, therefore, also 5.

Key concepts

Power Series

A power series is a series of the form \( \sum_{n=0}^{\infty} a_{n} x^{n} \). It's like a polynomial with potentially infinitely many terms. Here, each term of the series is made up of a coefficient \( a_{n} \) and a power of \( x^n \). Because power series can have infinitely many terms, instead of looking at the entire function at every possible \( x \), we focus on where it converges.
The series converges when, as you sum more terms, the total approaches a specific value. This stability in summation, depending on the value of \( x \), is what defines the series' radius of convergence.
Power series are incredibly useful in mathematics and engineering. They allow complicated functions to be represented in a simpler polynomial form, making them easier to manipulate and analyze over certain intervals.

Coefficients Growth

Coefficients in a power series, represented by \( a_n \), play a crucial role in determining the series' behavior, particularly its convergence properties. A "growth rate" refers to how these coefficients behave as \( n \) increases.
Consider a series \( \sum_{n=0}^{\infty} a_{n} x^{n} \). The convergence largely depends on how quickly or slowly the coefficients \( a_n \) grow or shrink. If the coefficients grow very rapidly, the terms of the series might also grow, impacting convergence.
For example, series with coefficients that grow proportionally to factorial functions tend to converge more slowly. In contrast, if the coefficients diminish, the series might converge quickly. The radius of convergence involves understanding where the series settles, and it remains identical even if the terms' signs change, since the alteration does not affect the coefficients growth rate.

Alternating Series

An alternating series is a type of series where the terms alternate in sign. The series \( \sum_{n=0}^{\infty}(-1)^{n} a_{n} x^{n} \) is an example. Here, multiplying each term by \((-1)^n\) creates the alternation because \((-1)^n\) results in either 1 or -1 as \( n \) changes.
This construction is significant in various mathematical applications as it can enhance convergence properties. It does not, however, affect the radius of convergence since the fundamental growth rate of the coefficients \( a_n \) remains unchanged.
Alternating series are particularly interesting because they can converge even when the corresponding non-alternating series do not. They add a level of elegance and complexity to analysis, which often proves beneficial for approximating functions.