11.5: Taylor Series
A power series is a series of the form
∞
X
an xn
n=0
where each an is a number and x is a variable.
P∞
A power series defines a function f (x) = n=0 an xn where we
substitute numbers for x.
P∞
Note: The function f is only defined for those x with n=0 an xn
convergent.
1
Geometric series as a power series
For |x| < 1 we computed
∞
X
xn =
n=0
2
1
1−x
Taylor Series
If f (x) is an infinitely differentiable function, then the Taylor series
of f (x) at a is the series
∞
X
f (n) (a)
(x − a)n
n!
n=0
3
Example
Compute the Taylor series of f (x) = ex at a = 0.
4
Solution
We know f (n) (x) = ex for all n ≥ 0. So f (n) (0) = 1 and the Taylor
series of f (x) = ex at a = 0 is
∞
X
1 n
x
n!
n=0
5
Example
Compute the Taylor series at a = 1 of f (x) =
6
√
x.
Solution
1
Write f (x) = x 2 . Then f 0 (x) = 12 x
, f 00 (x) =
−1 −3
2 ,
4 x
(−1)4+1 5·3·1 −7
−5 −7
2 =
x 2 . In general,
16 x
24
n+1
−2n−1
(−1)
(2n−1)·(2n−3)···3·1
x 2 so that
2n
n+1
(−1)
(2n−1)·(2n−3)···3·1
and the Taylor series of
2n
f 000 (x) = 38 x
f (n) (x) =
−1
2
−5
2
, f (4) (x) =
f (n) (1) =
√
f (x) = x at a = 1 is
1+
∞
X
n=1
(−1)n+1
(2n − 1)(2n − 3) · · · 3 · 1
(x − 1)n
n
n!2
7
Convergence of Taylor series
Given an infinitely differentiable function f (x) with Taylor series
P∞
P∞
(at a) n=0 bn (x − a)n either n=0 bn (x − a)n converges and is
equal to f (x) for every number x or there is a number R (called the
P∞
radius of convergence) for which n=0 bn (x − a)n converges and is
P∞
equal to f (x) for |x − a| < R while n=0 bn (x − a)n diverges for
|x − a| > R.
8
Examples
•
∞
X
1 n
e =
x
n!
n=0
x
for all x.
•
∞
X
1
=
xn
1 − x n=0
for |x| < 1 and diverges for |x| > 1
9
Operations on Taylor series
(n)
P∞
P∞
Differentiation: If f (x) = n=0 f n!(0) xn = n=0 an xn , then
(n+1)
P∞
P∞
f 0 (x) = n=0 f n! (0) xn = n=0 nan xn−1 .
P∞
Integration: If f (x) = n=0 an xn , then
R
P∞ an n+1
f (x)dx = C + n=0 n+1
x
.
P∞
P∞
Products: If f (x) = n=0 an xn and g(x) = n=0 bn xn , then
P∞ Pn
f (x) · g(x) = n=0 ( i=0 ai bn−i )xn .
P∞
Composition (monomial case): If f (x) = n=0 an xn and m is
P∞
a positive integer, then f (xm ) = n=0 an xnm .
10
Example
Compute the Taylor series at a = 0 of
R
2
ex dx.
11
Solution
P∞ 1 n
P∞ 1 2n
2
We know ex = n=0 n!
x . So, ex = n=0 n!
x . Integrating,
R x2
R P∞ 1 2n
P∞
1
2n+1
e dx =
.
n=0 n! x dx = C +
n=0 n!(2n+1) x
R x2
1 5
1 7
That is, e dx = C + x + 16 x3 + 30
x + 98
x + ···.
12
Example
Find the Taylor series at zero of
1+x2
1−x3 .
13
Solution
P∞ 3n
1
un so that 1−x
3 =
n=0 x .
P∞ 3n P∞
1+x2
2
Multiplying, 1−x
= n=0 (x3n + x3n+2 ) =
3 = (1 + x )
n=0 x
1 + x2 + x3 + x5 + x6 + x8 + x9 + x11 + · · ·
We know
1
1−u
=
P∞
n=0
14