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# Sarjakehitelmiä¶

$\begin{split}\begin{array}{cl}\hline \text{Sarjakehitelmä} & \text{Suppenemisväli} \\\hline & \\[-1ex] \dfrac{1}{1 - x} = \sum\limits_{k = 0}^{\infty}x^k = 1 + x + x^2 + x^3 + x^4 + \cdots & -1 < x < 1 \\[3ex] e^x = \sum\limits_{k = 0}^{\infty}\dfrac{x^k}{k!} = 1 + x + \dfrac{x^2}{2} + \dfrac{x^3}{6} + \dfrac{x^4}{24} + \cdots & \R \\[3ex] \sin x = \sum\limits_{k = 0}^{\infty}\dfrac{(-1)^{k}x^{2k + 1}}{(2k + 1)!} = x - \dfrac{x^3}{6} + \dfrac{x^5}{120} - \dfrac{x^7}{5040} + \dfrac{x^9}{362880} - \cdots & \R \\[3ex] \cos x = \sum\limits_{k = 0}^{\infty}\dfrac{(-1)^kx^{2k}}{(2k)!} = 1 - \dfrac{x^2}{2} + \dfrac{x^4}{24} - \dfrac{x^6}{720} + \dfrac{x^8}{40320} - \cdots & \R \\[3ex] \ln(1 + x) = \sum\limits_{k = 0}^{\infty}\dfrac{(-1)^kx^{k + 1}}{k + 1} = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dfrac{x^4}{4} + \dfrac{x^5}{5} - \cdots & -1 < x \leq 1 \\[3ex]\hline \end{array}\end{split}$
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