Term-by-term integration of a power series

Definition

Term-by-term integration of a power series involves integrating each term of the series individually within its interval of convergence. This process results in a new power series that is the integral of the original.

5 Must Know Facts For Your Next Test

1. If a power series $\sum a_n x^n$ converges on an interval, then its term-by-term integral $\int \sum a_n x^n dx = \sum \frac{a_n x^{n+1}}{n+1} + C$ also converges on that interval.
2. The constant of integration $C$ must be included when performing term-by-term integration.
3. The radius of convergence for the integrated series remains the same as that of the original power series.
4. Term-by-term differentiation and integration are valid operations within the radius of convergence.
5. Power series can be integrated term-by-term over any subinterval where they converge absolutely.

Review Questions

• What happens to the radius of convergence when you integrate a power series term by term?
• Describe how to integrate a given power series $\sum a_n x^n$ term by term.
• Why is it necessary to add the constant of integration when integrating a power series?

Related terms

Power Series: An infinite sum in the form $\sum_{n=0}^{\infty} a_n (x - c)^n$, where $a_n$ are coefficients and $c$ is the center.

Radius of Convergence: The distance from the center $c$ within which a power series converges.

Term-by-Term Differentiation: Differentiating each term in a power series individually, resulting in another power series.