Power Series

Definition

A power series is an infinite series of the form

$$\sum _{n=0}^{\infty }{c}_n{x}^n=c_0+c_{1}x+c_2{x}^2+c_3{x}^3+\dots$$

Intuition

A power series is a polynomial with infinitely many terms. Within its radius of convergence it behaves exactly like a smooth function — it can be differentiated and integrated term-by-term, making it a powerful tool for defining and computing with functions analytically.

Formal Description

The ratio test gives convergence when

$$|x|\underset{n\to \infty }{\lim }\left|\frac{c_{n+1}}{c_n}\right|<1⟺|x|<R,R=\underset{n\to \infty }{\lim }\left|\frac{c_n}{c_{n+1}}\right|.$$

$R$ is the radius of convergence. The series converges for $|x|<R$, diverges for $|x|>R$, and the boundary $|x|=R$ requires separate analysis.

Within $|x|<R$, the power series may be differentiated or integrated term-by-term:

$$\frac{d}{dx}\sum _{n=0}^{\infty }{c}_n{x}^n=\sum _{n=1}^{\infty }n{c}_n{x}^{n-1},$$ $$\int \left(\sum _{n=0}^{\infty }{c}_n{x}^n\right)dx=C+\sum _{n=0}^{\infty }\frac{c_n{x}^{n+1}}{n+1}.$$

Key Results

Example. The series $\sum _{n=0}^{\infty }\frac{x^n}{n!}$ has infinite radius of convergence ($R=\infty$) and is its own derivative — it equals $e^x$.

Applications

Power series are the analytic framework underlying Taylor Series. Differentiating and integrating known power series (e.g. the geometric series) is a standard technique for obtaining new series.

Trade-offs

Convergence at the boundary $|x|=R$ must be checked separately for each series; the ratio test is inconclusive there. Power series centered at $a=0$ may not converge for all $x$ of interest — shifting the center or using a different representation may be needed.

Links

• Ratio Test
• Geometric Series
• Taylor Series