FM for Actuaries

Review of Mathematics and Statistics 329
A.7 Integration by part
If u(x) and v(x) are functions of x,then
∫
∫
u(x) dv (x) =u(x)v(x) −
v(x) du(x).
This result applies to definite as well as indefinite integrals. Now if u(x) ≡ 1, we
have (as ∫ v(x) du(x) =0)
∫ b
a
dv(x) =v(b) − v(a).
A.8 Taylor series expansion
For a function y = f(x), the expansion of f(x +∆x) around x for a small value
of ∆x is, to the first order approximation, given by
df (x)
f(x +∆x) =f(x)+
dx (∆x)+o(∆x),
where
o(∆x)
∆x → 0
as ∆x → 0. The expansion to the second order approximation is
where
as ∆x → 0.
f(x +∆x) =f(x)+
df (x)
dx (∆x)+d2 f(x)
dx 2 (∆x) 2 + o((∆x) 2 ),
o((∆x) 2 )
(∆x) 2 → 0
A.9 Binomial expansion
For a positive integer n,
(x + y) n =
n∑
( n
i
i=0
)
x i y n−i ,
where
( n
i
)
=
n!
i!(n − i)! .