( )
k
k
f (m)
f
( ) (m)
n
k
k
x =
(x − m) =
(x − m) + R n
∑
∞
Πk
∑
n
Πk
k = 0
k =0
n
1
( )
Π n
n−k
k
=
(m) (x − m) + R n
∑ Πk Π(n − − k)
k = 0
∑
∞
∑
n
( )
k
k
−n
k
k
f (m)
f
( ) (m)
x =
(x − m) =
(x − m) + R n
Πk
Πk
k = 0
k k =0
n
Π(n − + k)
1
1
k
n
k
− −k
=
(−1)
(m)
(x − m) + R n n
∑ Πk
Π(n − 1)
k = 0
R n : Error after n terms

By ignoring terms higher than the fourth order, we get the following forms:

4
( )
n
1
Π n
n−k
k
x ≈
(m) (x − m)
(A2.18)
∑ Πk Π(n − k)
k =0
−
4
1
Π(n − + k)
n k
k
n
k
1
− −
x ≈ ∑ (−1)
(m)
(x − m)
(A2.19)
Πk
Π(n − 1)
k =0
After substituting Equations A2.18 and A2.19 into Equation A2.14, we
obtain the following results:
L n1 = −2C m
n
s
∞
1
⎛ (x − µ)
2
⎞
L 2 = C x
n
exp −
2
dx
n
∫
s
σ 2π
⎝ ⎜
2σ ⎠ ⎟
−∞
4
∞
1 Π( )
n
1
⎛ (x − µ)
2
⎞
n−k
k
≈ C s
(m) (x − m)
exp −
2
dx
∫ ∑ Πk Π( (n − k)
σ 2π
⎝ ⎜
2σ ⎠ ⎟
k =0
−∞
( −
⎧ n
n−1
n n 1) n−2
2
2 ⎫
m + n nm E X) − m) +
m ( ( ) − 2mE X
( (
E X
( ) + m )
⎪
⎪
2
⎪
⎪
⎪
(
n n − 1)(n − 2)
⎪
n− −3
3
2
2
3
+
m ( ( ) − 3mE X
( ) + 3m E X) − m
E X
(
)
⎪ ⎪
⎪ ⎪
6
= C s ⎨
⎬
⎪
⎪
n n
( − 1)(n − 2)(n − 3) n− 4
4
3
+
m ( ( ) − − 4mE X )
⎪
⎪
E X
(
24
⎪
⎪
⎪
⎪
2
2
3
4
⎪ + 6m E X ) − 4m E X
( ) + m
⎩
(
)
⎪ ⎭
Appendix 2: Product Upgrades Based on Minimum Expected Quality Loss 347