26 Power Series
26.1 Differentiating Term-By-Term
Uses Dominated Convergence for Derivatives
Proposition 26.1 (Differentiation & Radius of Convergence)
- Proof 1: assuming ratio test works
- Proof 2: general case (hard!)
Theorem 26.1 (Term-by-Term Differentiation)
Corollary 26.1 (Power Series are Smooth)
26.2 Power Series Representations
Definition 26.1 (Power Series Representation)
Remark: use a power series representation to generalize a function from real numbers to other objects (complex numbers, matrices).
Proposition 26.2 (Candidate Series Representation)
Definition 26.2 (Taylor Series)
26.2.1 Taylor’s Error Formula
Proposition 26.3 (A Generalized Rolle’s Theorem)
Proposition 26.4 (A Polynomial Mean Value Theorem)
Theorem 26.2 (Taylor Remainder)
26.2.2 Series Centered at \(a\in\mathbb{R}\)
Theorem 26.3 Formula for taylor series and error based at arbitrary \(a\in\RR\).
26.3 \(\bigstar\) Smooth vs Analytic
Remark: most of the time a taylor series converges to the desired function, but this is not required. Give example.
Definition 26.3 (Analytic Function)
Examples of analytic functions:
- Polynomials
- Exponential (to come)
- Sine, Cosine (to come)
These have power series that converge everywhere. But other examples exist
\[\frac{1}{1+x^2}\]
Give example of smooth but not analytic function!