Infinite series

Power series

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

𝑓(𝑥)=∞∑𝑛=0𝑏𝑛(𝑥−𝑎)𝑛

where (𝑏𝑛)∞𝑛=0 is an infinite sequence of real numbers, and 𝑎 ∈ℝ. In the case where 𝑎 =0 it is often just called a power series. The domain of 𝑓 is therefore the set 𝐷 ⊆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=lim𝑛→∞∣𝑏𝑛𝑏𝑛+1∣

where

• 𝑓(𝑥) is absolutely convergent on the interval centred at 𝑎 given by 𝑥 ∈(𝑎 −𝑅,𝑎 +𝑅) ∪{𝑎}
• 𝑓(𝑥) must be checked manually for divergence at each 𝑥 =𝑎 ±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 =∞, i.e. the limit diverges, then 𝑓(𝑥) is absolutely convergent for all 𝑥 ∈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 𝐶𝜔 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

𝑇𝑓∞,𝑎(𝑥)=𝑛∑𝑚=0𝑓(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.

---

tidy | SemBr