EC3070 FINANCIAL DERIATIVES
TAYLOR’S THEOREM AND SERIES EXPANSIONS
Taylor’s Theorem.
If
f
is a function continuous and
n
times differentiable in
an interval [
x, x
+
h
], then there exists some point in this interval, denoted by
x
+
λh
for some
λ
∈
[0
,
1], such that
f
(
x
+
h
) =
f
(
x
) +
hf
(
x
) +
h
2
2
f
(
x
) +
· · ·
· · ·
+
h
(
n
−
1)
(
n
−
1)!
f
(
n
−
1)
(
x
) +
h
n
n
!
f
n
(
x
+
λh
)
.
If
f
is a socalled analytic function of which the derivatives of all orders exist,
then one may consider increasing the value of
n
indefinitely.
Thus, if the
condition holds that
lim
n
→∞
h
n
n
!
f
n
(
x
) = 0
,
which is to say that the terms of the series converge to zero as their order
increases, then an infiniteorder Taylorseries expansion is available in the form
of
f
(
x
+
h
) =
∞
j
=0
h
j
j
!
f
j
(
x
)
.
This is obtained simply by extending indefinitely the expression from Taylor’s
Theorem. In interpreting the summary notation for the expansion, one must
be aware of the convention that 0! = 1.
A Taylorseries expansion is available for functions which are analytic
within a restricted domain. An example of such a function is (1
−
x
)
−
1
. The
function and its derivatives are undefined at the point
x
= 1. Nevertheless,
Taylorseries expansions exists for the function at all other points and for all

h

<
1. Another example is provided by the function log(
x
) which is defined
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 Spring '12
 D.S.G.Pollock
 Taylor Series, Taylor’s Theorem, D.S.G. POLLOCK, stephen pollock

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