Introduction to the Taylor expansion
Introduction to the Taylor expansion I
Remember that:
f ′ (a) = lim
h→0
We can approximate a point on a
curve at x = a + h by the
corresponding point on the tangent:
f (a + h) − f (a)
h
You can also think about it this way:
f ′ (a) =
f (a + h) − f (a)
+ terms
h
which goes to zero as h → 0
′
f (a + h) ≈ f (a) + hf (a)
For h close to 0, it is a good
approximation.
The terms which goes to zero as h → 0 have to be h × something and are
written O(h). More generally, we note:
Denition:
O(hn )
Introduction to the Taylor expansion II
So
f ′ (a) =
≡
terms of the form hn × something
Introduction to the Taylor expansion III
f (a + h) − f (a)
+ O(h)
h
Multiplying by h, we have :
Actually this expansion can be done with as many terms as we like:
h · f ′ (a) = f (a + h) − f (a) + O(h2 )
or we can write the following Taylor expansion of order 2:
f (a + h) = f (a) + h · f ′ (a) + O(h2 )
When h
Taylor's and Maclaurin's Expansions.
For a function f that has n continuous derivatives on the neighbourhood
of a, the taylor expansion of f is:
is small (e.g. h = 0.1), then h2 is even smaller (h2 = 0.01).
f (a + h) = f (a) + f ′ (a) h + f ′′ (a)
where f (k) is the k.th derivative of f .
called the Maclaurin's expansion.
Example:
Taylor's
Expansion
Exercise:
Example: Compute the rst three term of the Taylor expansion of the
function f (x) = ex at a = 2.
We have the derivatives ∀n, f
e2+h
(n)
(x) = e
x
, so the Taylor expansion becomes:
= e2 + e2 h + e2
2
2
=e +e h+
h2
2!
2
e2 h2
+ O(h3 )
3
+ O(h )
h2
hn
+ · · · + f (n) (a)
+ O(hn+1 )
2!
n!
When a = 0,
Taylor and Maclaurin expansions
Write down the Maclaurin expansion of:
1
sin(x)
2
cos(x)
3
tan(x)
4
sinh(x)
5
cosh(x)
this expansion is also
R
B
rook Taylor and Colin Maclaurin
emarks
The Taylor expansion gives you a nite expansion which approximate
the function.
A Taylor expansion is a nite series approximation of a function around
a point.
Now, we are going to extend this to innite series.
Brook Taylor
Colin Maclaurin
(1685-1731)
(1698-1746)
Exercise:
Taylor series
Taylor series
Taylor's formula with a remainder.
For a function f that has n continuous derivatives on the neighbourhood of a,
the taylor expansion of f is:
f (x) = f (a) + f ′ (a) (x − a) + f ′′ (a)
(x − a)2
(x − a)n
+ · · · + f (n) (a)
+ Rn (x, ξ)
2!
n!
with Rn (x, ξ) the remainder term dened as:
Rn (x, ξ)
= f (x) −
Pn
= f (n+1) (ξ)
k=0
f (k) (a)
(x−a)k
k!
(x−a)n+1
(n+1)!
assuming f (n+1) (ξ) is continuous on [a; x]. If limn→∞ Rn (x, ξ) = 0, then the
function f (x) can be written as an innite series:
f (x) =
∞
X
n=0
f (n) (a)
(x − a)n
n!
Compute the remainder term of the Maclaurin expansion of the following
function. Show that ∀x, this remainder is converging to 0. Write down then
the corresponding Taylor series.
1
ex
2
sin(x)