1
Power Series and Euler’s Formula
Power Series and Euler’s Formula
At x D 0, the nth derivative of x n is the number n!
Other derivatives are 0:
Multiply the nth derivatives of f .x/ by x n =nŠ to match function with series
xn
x
x2
f .x/ D f .0/ C f � .0/ C f �
.0/ C ��� C f .n/ .0/ C ���
1
2
nŠ
TAYLOR
SERIES
f .x/ D e x
EXAMPLE 1
Match with x n =n !
All derivatives D 1 at x D 0
xn
x
x2
Taylor Series
D e x D 1 C 1 C 1 C ��� C 1 C ���
Exponential Series
1
2
nŠ
f D sin x
EXAMPLE 2
At x D 0 this is 0
sin x D 1 �
�1
0
x
x3
x5
�
1 C 1 ����
1
3Š
5Š
cos x D 1 � 1
x2
x4
C 1 ����
2Š
4Š
f � D � sin x
1 0
�1
REPEAT
0
�1
0 1
EVEN POWERS
i 2 D �1 and then i 3 D �i
�1
0
REPEAT
Find the exponential e ix
Those are
cos x C i sin x
e ix D cos x C i sin x
EULER’S GREAT FORMULA
ei 0
d
.cos x/ D � sin x
dx
1
1
e ix D 1 C ix C .ix/2 C .ix/3 C ���
2Š 3Š
x3
x2
D 1�
C ��� C i x �
C ���
2Š
3Š
e i
i sin f � D � cos x
ODD POWERS sin.�x/ D � sin x
f D cos x produces 1
EXAMPLE 3
Imaginary
1 0
f � D cos x
cos e i D cos C i sin Real
part
e i C e �i D 2 cos e i D �1 combines 4 great numbers
Two more examples of Power Series (Taylor Series for f .x/)
f .x/ D
1
1�x
D 1 C x C x 2 C x 3 C ���
f .x/ D � ln.1 � x/ D
“Geometric series”
x x2 x3 x4
C
C
C
C ���
1
2
3
4
“Integral of geometric series”
MIT OpenCourseWare
http://ocw.mit.edu
Resource: Highlights of Calculus
Gilbert Strang
The following may not correspond to a particular course on MIT OpenCourseWare, but has been
provided by the author as an individual learning resource.
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.