Interval of Convergence

Definition

The interval of convergence is the range of values of the independent variable for which a power series converges, or in other words, the set of values where the series represents a function. This concept is central to understanding the properties and applications of power series, Taylor series, and Maclaurin series.

5 Must Know Facts For Your Next Test

1. The interval of convergence is always a symmetric interval centered at the point of expansion, which is typically 0 for Maclaurin series.
2. The interval of convergence can be determined using the ratio test or the root test, which examine the behavior of the coefficients in the power series.
3. Power series have the property of term-by-term differentiation and integration within their interval of convergence.
4. The interval of convergence is crucial for determining the range of values over which a Taylor or Maclaurin series approximation is valid.
5. Knowing the interval of convergence allows you to use power series to represent and approximate functions, which is a powerful tool in calculus.

Review Questions

• Explain the relationship between the interval of convergence and the properties of power series.
  • The interval of convergence is directly linked to the properties of power series, as it determines the range of values over which the series converges and represents a function. Within the interval of convergence, the power series can be differentiated and integrated term-by-term, and it can be used to approximate the function. Understanding the interval of convergence is essential for applying the powerful tools of power series, Taylor series, and Maclaurin series in calculus.
• Describe how the interval of convergence is determined for a power series.
  • The interval of convergence of a power series is typically determined using the ratio test or the root test. The ratio test examines the behavior of the coefficients in the series, looking for a limit that will define the radius of convergence. The root test looks at the nth root of the coefficients, again seeking a limit that will establish the radius of convergence. Once the radius of convergence is known, the interval of convergence can be defined as the symmetric interval centered at the point of expansion, usually 0 for Maclaurin series.
• Analyze the importance of the interval of convergence in the context of Taylor and Maclaurin series.
  • The interval of convergence is crucial for determining the range of values over which a Taylor or Maclaurin series approximation is valid. These series are used to represent and approximate functions, and knowing the interval of convergence allows you to understand the limitations of the approximation. Within the interval of convergence, the series can be used to accurately represent the function and perform operations like differentiation and integration. Outside the interval of convergence, the series diverges and cannot be used to represent the function. Therefore, the interval of convergence is a fundamental concept in understanding the power and limitations of Taylor and Maclaurin series in calculus.

Related terms

Power Series: A power series is an infinite series where each term is a constant multiplied by the independent variable raised to a non-negative integer power.

Convergence: Convergence refers to the behavior of a series, where the sequence of partial sums approaches a finite limit as the number of terms increases.

Radius of Convergence: The radius of convergence is the distance from the center of a power series to the nearest point where the series ceases to converge.