> [!definition] > > Let $\cx$ be a [[Complex Numbers|complex]] [[Banach Algebra|Banach algebra]] and $\bracs{c_n}_0^\infty \subset \mathbb{C}$. The **power series** of $c_n$ about $a \in \cx$ is the function > $ > f(x) = \sum_{n = 0}^\infty c_n(x - a)^n = \limv{n}\sum_{k = 0}^n c_k(x - a)^k > $ > defined on the points on which it exists, where the $n$-th **partial sum** of $f$ is > $ > f_n(x) = \sum_{k = 0}^n c_k(x - a)^k > $ > [!definition] > > Let $\cx$ be a unital Banach algebra, $\bracs{c_n}_0^\infty \subset \complex$, and $f = \sum_{n = 0}^\infty c_n(x - a)^n$ be the power series of $\bracs{c_n}_0^\infty$ about $a$, let > $ > C = \limsup_{n \to \infty} \abs{c_n}^{1/n} > $ > then $R = C^{-1}$ is the **radius of convergence**, where the power series of $\bracs{c_n}_0^\infty$ converges absolutely for all $x \in \cx$ with $\norm{x - a} < R$, and the series converges as a function [[Uniform Convergence|uniformly]] on any radius strictly less than $R$. > > *Note: divergence outside of $R$ is not guaranteed.* > > *Proof*. Let $x \in B(a, R)$, then > $ > \limsup_{n \to \infty}\abs{c_n}^{1/n} < \sup_{x \in B(a, R)}\norm{x - a}^{-1} > $ > so $\abs{c_n}^{1/n} < \norm{x - a}^{-1}$ eventually. Let $L \in (C, \norm{x - a}^{-1})$ and $N \in \nat$ such that $\abs{c_n}^{1/n} \le L$ for all $n \ge N$, giving > $ > \begin{align*} > \abs{c_n} &\le L^n < \norm{x - a}^{-n} \\ > \abs{c_n} \cdot \norm{x - a}^n &\le \braks{\frac{L}{\norm{x - a}}}^{n} > \end{align*} > $ > for all $n \ge N$ with $L/\norm{x - a} < 1$, so > $ > \begin{align*} > \sum_{n \in \nat_0}\norm{c_n (x - a)^n} &\le \sum_{n \in \nat_0}\abs{c_n} \cdot \norm{x - a}^n \\ > &= \sum_{n < L}\abs{c_n} \cdot \norm{x - a}^n + \sum_{n \ge L}\abs{c_n} \cdot \norm{x - a}^n \\ > &\le \sum_{n < L}\abs{c_n} \cdot \norm{x - a}^n + \sum_{n \ge L}\abs{c_n} \cdot \braks{\frac{L}{\norm{x - a}}}^n > \end{align*} > $ > which converges absolutely. As this holds for all points $x \in B(a, R)$, the sum converges uniformly as a function. > [!theorem] > > Let $\cx$ be a Banach algebra, $\bracs{c_n}_0^\infty \subset \mathbb{C}$, and $f(x) = \sum_{n = 0}^\infty c_n(x - a)^n$ be a power series of $\bracs{c_n}_0^\infty$ around $a$, then its formal derivative > $ > g(x) = \sum_{n = 1}^\infty nc_n(x - a)^{n - 1} > $ > has the same radius of convergence as $f$. Therefore the series converges uniformly, and the [[Derivative|derivative]] can be taken term-wise. > > *Proof*. Since $\abs{n}^{1/n} \to 1$ is convergent and $\seq{\abs{c_n}}$ is bounded, > $ > C = \limsup_{n \to \infty}\abs{nc_n}^{1/n} = \limv{n}\abs{n}^{1/n} \cdot \limsup_{n \to \infty}\abs{c_n}^{1/n} = \limsup_{n \to \infty}\abs{c_n}^{1/n} > $ > From the power rule, the size of each term's derivative may be bounded as > $ > \begin{align*} > \norm{D(x - a)^n(x_0)(h)} &\le \sum_{k = 1}^n\norm{x_0 - a}^{k - 1} \cdot \norm{h} \cdot \norm{x_0 - a}^{n - k} \\ > &= n\norm{x_0 - a}^{n} \cdot \norm{h} \\ > \norm{D(x - a)^n(x_0)} &\le n\norm{x_0 - a}^{n} \\ > \norm{c_nD(x - a)^n(x_0)} &\le \abs{nc_n} \cdot \norm{x_0 - a}^n > \end{align*} > $ > By the same argument, we have > $ > \sum_{n = 0}^\infty \abs{c_n} \cdot \norm{D(x - a)^n(x_0)} \le \sum_{n = 0}^{\infty}\abs{nc_n} \cdot \norm{x_0 - a}^n > $ > where the latter converges absolutely. > > Since the derivatives of the partial sums agree with the partial sums of the derivatives and the series of $g$ converges uniformly, the limit can be interchanged with the derivative, giving $Df = g$. > [!theorem] > > Let $\alg$ be a Banach algebra, then the [[Power Series|power series]] > $ > x \mapsto \sum_{n = 0}^\infty c_nx^n > $ > is $C_\infty$ in its radius of convergence. > > *Proof*. Since the derivative of each power series is another power series with the same radius of convergence, all power series are infinitely continuously differentiable in their radius of convergence. > [!theorem] Geometric Series > > The [[Geometric Series|geometric series]] allows the conversion of its converging formula into a power series > $ > \frac{1}{1 - x} = \sum_{n = 0}^{\infty}x^{n} > $ > that converges for $|x| \lt 1$, which can be written as $|x| \lt R$ where $R$ is the radius of convergence. [^1]: Past a certain point all the terms become big and the series explodes.