power series — Jemoka Knowledge Base

a power series centered at a is defined with c_{n} \in \mathbb{R}, whereby:

\begin{equation} f(x) = \sum_{n=0}^{\infty} c_{n}(x-a)^{n} \end{equation}

meaning it is written as c_0 + c_1(x-a) + c_2(x-a)^{2} + c_3 (x-a)^{3} + \cdots radius of convergence there is a radius of convergence R \geq 0 for any power series, possibly infinite, by which the series is absolutely convergent where |x-a| < R, and it does not converge when |x-a| > R , the case where |x-a| = R is uncertain ratio test: if all coefficients c_{n} are nonzero, and some \lim_{n \to \infty} \left| \frac{c_{n}}{c_{n+1}} \right| evaluates to some c — if c is positive or +\infty, then that limit is equivalent to the radius of convergence Taylor’s Formula: a power series f(x) can be differentiated, integrated on the bounds of (a-R, a+R), the derivatives and integrals will have radius of convergence R and c_{n} = \frac{f^{(n)}(a)}{n!} to construct the series linear combinations of power series When \sum_{n=0}^{\infty} a_{n} and \sum_{n=0}^{\infty} b_{n} are both convergent, linear combinations of them can be described in the usual fashion:

\begin{equation} c_1 \sum_{n=0}^{\infty} a_{n}+ c_2 \sum_{n=0}^{\infty} b_{n} = \sum_{n=0}^{\infty} c_1 a_{n} + c_2 b_{n} \end{equation}

some power series geometric series \begin{equation} 1 + r + r^{2} + r^{3} + \dots = \sum_{n=0}^{\infty} r^{n} = \frac{1}{1-r} \end{equation} which converges -1 < r < 1, and diverges otherwise. bounded geometric sum \begin{equation} \sum_{t=0}^{n} x^{t} = \frac{x^{n+1} -1}{x-1} \end{equation} for x \neq 1. exponential series \begin{equation} 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots = \sum_{n=0}^{\infty} \frac{x^{n}}{n!} = e^{x} \end{equation} which converges for all x \in \mathbb{R}. absolutely convergent If:

\begin{equation} \sum_{n=0}^{\infty} |a_{n}| \end{equation}

converges, then:

\begin{equation} \sum_{n=0}^{\infty} a_{n} \end{equation}

also converges. This situation is called absolutely convergent.