1Differentiation
IA Differential Equations
1.4 Taylor’s theorem
Theorem (Taylor’s Theorem). For n-times differentiable f, we have
f(x + h) = f(x) + hf
0
(x) +
h
2
2!
f
00
(x) + ··· +
h
n
n!
f
(n)
(x) + E
n
,
where E
n
= o(h
n
) as h → 0. If f
(n+1)
exists, then E
n
= O(h
n+1
).
Note that this only gives a local approximation around
x
. This does not
necessarily tell anything about values of f far from x (but sometimes does).
An alternative form of the sum above is:
f(x) = f (x
0
) + (x − x
0
)f
0
(x
0
) + ··· +
(x − x
0
)
n
n!
f
(n)
(x
0
) + E
n
.
When the limit as
n → ∞
is taken, the Taylor series of
f
(
x
) about the point
x = x
0
is obtained.