Power series

Definition

A power series is an infinite series of the form $\sum_{n=0}^{\infty} a_n (x - c)^n$, where $a_n$ represents the coefficient of the nth term and $c$ is a constant. Power series can be used to represent functions within their interval of convergence.

5 Must Know Facts For Your Next Test

1. The radius of convergence is determined using the ratio test or root test.
2. Within the interval of convergence, a power series converges absolutely and uniformly.
3. Power series can be differentiated and integrated term-by-term within their interval of convergence.
4. If two power series are equal on an interval, then their coefficients must be identical for all terms in that interval.
5. Common functions such as $e^x$, $\sin(x)$, and $\cos(x)$ can be represented as power series.

Review Questions

• What is the general form of a power series?
• How do you determine the radius and interval of convergence for a power series?
• Explain how differentiation and integration apply to power series within their interval of convergence.

Related terms

Radius of Convergence:

The distance from the center $c$ within which a power series converges.

Taylor Series:

A special type of power series generated by expanding a function about a point $a$. It takes the form $\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$.

Maclaurin Series:

$A Taylor Series centered at zero ($a = 0$). It has the form \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$.$