Noun
Maclaurin series (plural Maclaurin series)
(calculus) Any Taylor series that is centred at 0 (i.e., for which the origin is the reference point used to derive the series from its associated function); for a given infinitely differentiable complex function
f
{\displaystyle \textstyle f}
, the power series
f
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1
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3
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⋯
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∑
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∞
f
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n
{\displaystyle \textstyle f(0)+{\frac {f'(0)}{1!}}x+{\frac {f''(0)}{2!}}x^{2}+{\frac {f'''(0)}{3!}}x^{3}+\cdots =\sum _{n=0}^{\infty }{\frac {f^{(n)}(0)}{n!}}\,x^{n}}
.
Examples The Maclaurin series for any polynomial is the polynomial itself. Source: Internet
Setting 0 as the start of computation we get the simplified Maclaurin series : The same method of calculating the initial values from the coefficients can be used as for polynomial functions. Source: Internet
Using now the power series definition from above we see that for real values of x : In the last step we have simply recognized the Maclaurin series for cos(x) and sin(x). Source: Internet