5.1 Review of Power Series. Definition: Σ_{n=0}^{∞}a_{n}(x − x_{0})^{n} = lim_{n→∞}Σ_{n=0}^{k}a_{n}(x − x_{0})^{n} Taylor’s Theorem Suppose f has n + 1 continuous derivatives on an open interval containing a. Then for each x in the interval,

f(x) = [∑_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x − a)^{k}] + R_{n+1}(x)

where the error term R_{n+1}(x) satisfies R_{n+1}(x) = \frac{f^{(n+1)}(c)}{(n + 1)!}(x − a)^{n+1} for some c between a and x.

The infinite Taylor series converges to f,

f(x) = ∑_{k=0}^{∞} \frac{f^{(k)}(a)}{k!}(x − a)^{k} if and only if lim_{n→∞} R_{n}(x) = 0.

Defn: The function f is said to be analytic at a is its Taylor series expansion about x = a has a positive radius of convergence. 1.) Σ_{n=0}^{∞}a_{n}(x − x_{0})^{n} converges at the point x if and only if lim_{n→∞}Σ_{n=0}^{k}a_{n}(x − x_{0})^{n} exists at the point x. 2.) Σ_{n=0}^{∞}a_{n}(x−x_{0})^{n} converges absolutely at the point x if and only if Σ_{n=0}^{∞}|a_{n}||x−x_{0}|^{n} converges at the point x If a series converges absolutely, then it also converges. 3.) Ratio test for absolute convergence: Let L = lim_{n→∞}|\frac{a_{n+1}}{a_{n}} | lim_{n→∞}|\frac{a_{n+1}(x−x_{0})^{n+1}}{a_{n}(x−x_{0})^{n}} | = |x − x_{0}|lim_{n→∞}|\frac{a_{n+1}}{a_{n}} | = |x − x_{0}|L The power series converges at the value x if |x − x_{0}| < \frac{1}{L} The power series diverges at the value x if |x − x_{0}| > \frac{1}{L} The ratio test give no info at the value x if |x − x_{0}| = \frac{1}{L} Note \frac{1}{L} is the radius of convergence.