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Last modified: January 28, 2020
Taylor and Maclaurin Series
---------------------------
A Taylor series is a representation of a function
as an infinite sum of terms that are calculated
from the values of the function's derivatives at
a single point. It is common practice to approximate
a function by using a finite number of terms of its
Taylor series.
f(x) = f(a) + f'(a)(x - a) + f''(a)(x - a)2/2!
+ f'''(a)(x - a)3/3!...
= Σfn(x - a)n/n!
n
If the Taylor series is centered at zero, then
that series is also called a Maclaurin series.
f(x) = f(0) + f'(a)(x) + f''(a)(x)2/2!
+ f'''(a)(x)3/3!...
= Σfn(x)n/n!
n
Ex:
cos(x) = cos(0) - xsin(0) - x2/2cos(0) ...
= 1 - x2/2! + x4/4!
sin(x) = sin(0) + xcos(0) - x2/2sin(0) ...
= x - x3/3! + x5/5!
ex = 1 + x + x2/2 + x3/6 ...
Euler's formula:
eix = 1 + ix - x2/2! - ix3/3! ...
= (1 - x2/2! ...) + i(x - x3/3! ...)
= cos(x) + isin(x)
The Taylor series for 2 variables is:
f(x,y) = f(a,b) + (x - a)∂f/∂x + (y - b)∂f/∂y
+ (1/2)(x - a)2∂2f/∂x2
+ (x - a)(y - b)∂2f/∂x∂y
+ (1/2)(y - b)2∂2f/∂y2]
f(x,y) = f(0,0) + s∂f/∂s + t∂f/∂t
+ (1/2)s2∂2f/∂s2
+ st∂2f/∂s∂t
+ (1/2)t2∂2f/∂t2]