Power series

A power series centred at $𝑎$ is an infinite series with variable terms of the form

𝑓 (𝑥) = \infty \sum 𝑛 = 0 𝑏 𝑛 (𝑥 - 𝑎) 𝑛

where $(𝑏 𝑛) \infty 𝑛 = 0$ is an infinite sequence of real numbers, and $𝑎 \in ℝ$ . In the case where $𝑎 = 0$ it is often just called a power series. The domain of $𝑓$ is therefore the set $𝐷 \subseteq ℝ$ for which the series converges. A variant of the Ratio test for absolute convergence can be used to find the radius of convergence

𝑅 = l i m 𝑛 \to \infty ∣ 𝑏 𝑛 𝑏 𝑛 + 1 ∣

where

$𝑓 (𝑥)$ is absolutely convergent on the interval centred at $𝑎$ given by $𝑥 \in (𝑎 - 𝑅, 𝑎 + 𝑅) \cup {𝑎}$
$𝑓 (𝑥)$ must be checked manually for divergence at each $𝑥 = 𝑎 \pm 𝑅$
$𝑓 (𝑥)$ is divergent for all other $𝑥$

Note this gives rise to the special cases

If $𝑅 = 0$ then $𝑓 (𝑥)$ is absolutely convergent for $𝑥 = 𝑎$ and divergent everywhere else.
If $𝑅 = \infty$ , i.e. the limit diverges, then $𝑓 (𝑥)$ is absolutely convergent for all $𝑥 \in ℝ$ .

Another property arising from this is that it is not possible for a power series to be convergent in several separate points or intervals.

Power series are used to define the notion of an analytic function, i.e. the $𝐶 𝜔$ differentiability class. Note the derivative of a power series always has the same radius of convergence.

Examples

Perhaps the most well known power series is the Taylor series centred at $𝑎$ , often denoted

𝑇 𝑓 \infty, 𝑎 (𝑥) = 𝑛 \sum 𝑚 = 0 𝑓 (𝑚) (𝑎) 𝑚! (𝑥 - 𝑎) 𝑚

which in the case of $𝑎 = 0$ is called the Maclaurin polynomial. In fact, Borel's theorem states that every power series is in fact a Taylor series of some smooth function.

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