Maclaurin series
Noun
Meaning
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
(
0
)
+
f
′
(
0
)
1
!
x
+
f
″
(
0
)
2
!
x
2
+
f
‴
(
0
)
3
!
x
3
+
⋯
=
∑
n
=
0
∞
f
(
n
)
(
0
)
n
!
x
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
Examples The Maclaurin series for any polynomial is the polynomial itself.
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.
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).