Interval of convergence

Definition

The interval of convergence refers to the set of values for which a power series converges to a finite limit. It is crucial in understanding where a series can be used to represent functions accurately, as it dictates the domain over which the power series is valid. The endpoints of this interval may or may not be included, depending on whether the series converges at those points.

5 Must Know Facts For Your Next Test

1. To determine the interval of convergence, one typically uses the ratio test or the root test to find the radius of convergence, $$R$$.
2. The interval of convergence can be an open interval, closed interval, or half-open interval, depending on the behavior of the power series at its endpoints.
3. When testing convergence at the endpoints, it's important to analyze each endpoint separately, as they can behave differently.
4. The concept of interval of convergence is essential for representing functions with power series, especially in defining Taylor and Maclaurin series.
5. Power series with different centers will have different intervals of convergence; knowing one does not imply knowledge about another.

Review Questions

• How do you determine the interval of convergence for a given power series?
  • To determine the interval of convergence for a power series, you first apply either the ratio test or root test to find the radius of convergence, $$R$$. This gives you a center point around which you can construct an interval. You then check the endpoints by substituting them back into the original power series to see if they converge or diverge, resulting in a final interval that may be open, closed, or half-open.
• What role does the radius of convergence play in finding the interval of convergence?
  • The radius of convergence provides a key measurement that defines how far from the center point a power series converges. Once you find $$R$$ using tests like the ratio test, you can establish an initial interval as $$ (c - R, c + R) $$. However, this is just a starting point; you must still check both endpoints to see if they belong to the interval based on their individual convergence characteristics.
• Evaluate how understanding intervals of convergence impacts your ability to use Taylor and Maclaurin series effectively.
  • Understanding intervals of convergence is vital for effectively utilizing Taylor and Maclaurin series because it dictates where these approximations are valid. If a Taylor series converges within a certain interval but diverges outside it, you cannot confidently use it for values outside that range. Additionally, recognizing how function behavior varies near endpoints helps avoid errors when applying these series for function approximation and analysis.

Related terms

power series: A power series is an infinite series of the form $$ ext{a}_0 + ext{a}_1(x - c) + ext{a}_2(x - c)^2 + ...$$ where $$ ext{a}_n$$ are coefficients and $$c$$ is the center of the series.

radius of convergence: The radius of convergence is the distance from the center of a power series within which the series converges, often denoted as $$R$$.

absolute convergence: A series is said to be absolutely convergent if the series formed by taking the absolute values of its terms also converges.