Hva er Maclaurin-rekken?
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Start med den kjente rekken eu=∑n=0∞unn!\displaystyle e^u = \sum_{n=0}^{\infty}\frac{u^n}{n!}eu=n=0∑∞n!un
Sett u=−x2u = -x^2u=−x2:
e−x2=∑n=0∞(−x2)nn!=∑n=0∞(−1)nx2nn!\displaystyle e^{-x^2} = \sum_{n=0}^{\infty}\frac{(-x^2)^n}{n!} = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{n!}e−x2=n=0∑∞n!(−x2)n=n=0∑∞n!(−1)nx2n
=1−x2+x42−x66+⋯\displaystyle = 1 - x^2 + \frac{x^4}{2} - \frac{x^6}{6} + \cdots=1−x2+2x4−6x6+⋯
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