Term-by-term differentiation of a power series

Definition

Term-by-term differentiation of a power series involves differentiating each term of the series individually. This process is valid within the radius of convergence of the original power series.

5 Must Know Facts For Your Next Test

1. The radius of convergence for the differentiated series is the same as that of the original power series.
2. Within its radius of convergence, a power series can be differentiated term-by-term to produce another power series.
3. If $\sum_{n=0}^{\infty} c_n x^n$ converges for $|x| < R$, then $\sum_{n=1}^{\infty} n c_n x^{n-1}$ also converges for $|x| < R$.
4. Term-by-term differentiation simplifies finding derivatives of functions represented by power series.
5. The boundary behavior (at $|x| = R$) may differ between the original and differentiated series.

Review Questions

Related terms

Power Series:

An infinite sum in the form $\sum_{n=0}^{\infty} c_n (x - a)^n$, where $c_n$ are coefficients and $(x - a)$ is raised to increasing powers.

Radius of Convergence:

The distance from the center of convergence within which a power series converges.

Uniform Convergence:

A type of convergence where a sequence of functions converges to a limiting function uniformly on every interval in their domain.