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@jemoka / Jemoka Knowledge Base / wiki/concepts/power_series_o.md
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--- title: "power series" type: concept related: [Linear Combination, Power Series O] source: https://www.jemoka.com/posts/kbhpower_series_o/ confidence: high status: active --- 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.