Lecture 25Section 11.5 Taylor Polynomials in x; Taylor
Series in x
Jiwen He
1
1.1
Taylor Polynomials
Taylor Polynomials
Taylor Polynomials
Taylor Polynomials
The nth Taylor polynomial at 0 for a function f is
00
f (n) (0) n
f (0) 2
x + ··· +
x ;
Pn (x) = f (0) + f 0 (0)x +
2!
n!
Pn is the polynomial that has the same value as f at 0 and the same first n
derivatives:
00
00
Pn (0) = f (0), Pn0 (0) = f 0 (0), Pn (0) = f (0), · · · , Pn(n) (0) = f (n) (0).
Best Approximation
Pn provides the best local approximation of f (x) near 0 by a polynomial of degree
≤ n.
P0 (x) = f (0),
P1 (x) = f (0) + f 0 (0)x,
00
f (0) 2
P2 (x) = f (0) + f (0)x +
x .
2!
0
Taylor Polynomials of the Exponential f (x) = ex
00
f (n) (0) n
f (0) 2
x + ··· +
x ;
Pn (x) = f (0) + f 0 (0)x +00
x
(n)
2! x
n!
f (x) = ex , f 0 (x)
=
e
,
f
·
·
·
,
f
(x)
= ex ;
00 (x) = e ,
f (0) = 1, f 0 (0) = 1, f (0) = 1, · · · , f (n) (0) = 1.
1
Taylor Polynomials of f (x) = ex
P0 (x) = 1,
P1 (x) = 1 + x,
x2
,
2!
2
x
x3
+ ,
P3 (x) = 1 + x +
2!
3!
..
.
2
x
x3
xn
Pn (x) = 1 + x +
+
+ ··· +
.
2!
3!
n!
P2 (x) = 1 + x +
Taylor Polynomials of the Sine f (x) = sin x
2
00
f (0) 2
f (n) (0) n
000
P
(x)
=
x
+
·
·
·
+
0 f (0) + f 00(0)x +
n
f (x) = sin x, f0 (x) = cos x, f (x) 2!
=000− sin x, f (x) = −xcos; x, · · · ;
f (0) = 0, f (0) = 1, f (0) = 0, f (0) = −1, f (4)n!
(0) = 0, · · · ;
0
00
Taylor Polynomials of f (x) = sin x
P0 (x) = 0,
P1 (x) = P2 (x) = x,
x3
,
3!
x3
+
P5 (x) = P6 (x) = x −
3!
x3
P7 (x) = P8 (x) = x −
+
3!
P3 (x) = P4 (x) = x −
1.2
x5
,
5!
x5
x7
− ,
5!
7!
Remainder Term
Remainder Term
Remainder Term
Define the nth remainder by Rn (x) = f (x) − Pn (x); that is f (x) = Pn (x) +
Rn (x). Then
lim Pn (x) = f (x) if and only if
lim Rn (x) = 0
n→∞
n→∞
Taylor’s Theorem
If f has n + 1 continuous derivatives on an open interval I that contains 0, then
Z
for each x ∈ I,
1 x (n+1)
Rn (x) =
f
(t)(x − t)n dt.
n!
0
Lagrange Formula for the Remainder
For some number c between 0 and x,
f (n+1) (c) n+1
Rn (x) =
x
.
(n + 1)!
3
Taylor Polynomials of the Exponential f (x) = ex
f (n) (0) n
f (n+1) (c) n+1
Pn (x) = f (0) + f 0 (0)x + · · · +
x ; Rn (x) =
x
.
n!
(n + 1)!
x
Taylor Polynomials of the Exponential f (x) = e
x2
x3
xn
f (x) = ex , Pn (x) = 1 + x +
+
+ ··· +
.
2!
3!
n!
Remainder Term
For each real x, Rn (x) → 0 as n → ∞.
Proof.
Let J be the interval that joins 0 to x and let M = max et .
t∈J
f (n+1) (t) = et for all n, then max |f (n+1) (t)| = M.
t∈J
|x|n+1
→ 0 as
|Rn (x)| ≤ M
(n + 1)!
Note that
n → ∞.
Taylor Polynomials of the Sine f (n)
(x) = sin x
f (0) n
f (n+1) (c) n+1
n+1
Pn (x) = f (0) + f 0 (0)x +(n+1)
··· +
x ; Rn (x) =
x
.
|x|
1)!0].
, J = [0, x](nor+[x,
|Rn (x)| ≤ max |f
(t)| n!
t∈J
(n + 1)!
Taylor Polynomials of the Sine f (x) = sin x
x3
x5
x7
f (x) = sin x, P7 (x) = P8 (x) = x −
+
− , and so on.
3!
5!
7!
Remainder Term
For each real x, Rn (x) → 0 as n → ∞.
∀k, f (k) (t) = ± cos t or ± sin t, then max |f (n+1) (t)| ≤ 1.
t∈J
|x|n+1
→ 0 as n → ∞.
|Rn (x)| ≤
(n + 1)!
2
2.1
Taylor Series
Taylor Series
Taylor Series
Taylor Polynomial and the Remainder
If f (x) is infinitely differentiable on interval I containing 0, then
f (x) = Pn (x) + Rn (x), ∀x ∈ I;
n
(k)
X
f (0) k
f (n) (0) n
Pn (x) =
x = f (0) + f 0 (0)x + · · · +
x ,
k!
n!
k=0
Z
f (n+1) (c) n+1
1 x (n+1)
Rn (x) =
f
(t)(x − t)n dt.
x
or Rn (x) =
(n + 1)!
n! 0
Taylor Series
n
X
f (k) (0)
xk → f (x). [1ex] In this case,
k!
k=0
f (x) can be expanded as a Taylor series in x and write
∞
X
f (k) (0) k
f (x) =
x .
k!
If Rn (x) → 0 as n → ∞, then Pn (x) =
k=0
4
Taylor Series of the Exponential f (x) = ex
f (x) = Pn (x) + Rn (x),
Pn (x) =
n
X
f (k) (0)
k!
k=0
If lim Rn (x) → 0, then f (x) =
n→∞
∞
X
f (k) (0)
k!
k=0
xk
xk = lim Pn (x).
n→∞
Taylor Series of the Exponential f (x) = ex
∞
X
xk
x2
x3
ex =
=1+x+
+
+ ···
k!
2!
3!
for all real x
k=0
Number e
e=
∞
X
1
1
1
= 1 + 1 + + + ···
k!
2! 3!
k=0
Taylor Series of the Sine f (x) = sin x
f (x) = Pn (x) + Rn (x),
Pn (x) =
n
X
f (k) (0)
k=0
If lim Rn (x) → 0, then f (x) =
n→∞
∞
X
k=0
k!
xk
f (k) (0) k
x = lim Pn (x).
n→∞
k!
Taylor Series of the Sine f (x) = sin x
∞
X
x3
x5
x7
(−1)k 2k+1
x
=x−
+
−
+ ···
sin x =
(2k + 1)!
3!
5!
7!
for all real x
k=0
Number sin 1
sin 1 =
∞
X
1
1
1
(−1)k
= 1 − + − + ···
(2k + 1)!
3! 5! 7!
k=0
Taylor Series of the Cosine f (x) = cos x
f (x) = Pn (x) + Rn (x),
Pn (x) =
n
X
f (k) (0)
k=0
If lim Rn (x) → 0, then f (x) =
n→∞
∞
X
k=0
k!
xk
f (k) (0) k
x = lim Pn (x).
n→∞
k!
Taylor Series of the Cosine f (x) = cos x
∞
X
(−1)k 2k
x2
x4
x6
cos x =
x =1−
+
−
+ ···
(2k)!
2!
4!
6!
for all real x
k=0
Number cos 1
cos 1 =
∞
X
(−1)k
k=0
(2k)!
=1−
5
1
1
1
+ − + ···
2! 4! 6!
Taylor Series of the Logarithm f (x) = ln(1 + x)
n
X
f (k) (0) k
f (x) = Pn (x) + Rn (x), Pn (x) =
x
k!
k=0
If lim Rn (x) → 0, then f (x) =
n→∞
∞
X
f (k) (0)
k!
k=0
xk = lim Pn (x).
n→∞
Taylor Series of the Logarithm f (x) = ln(1 + x)
∞
X
(−1)k+1 k
x2
x3
x4
x =x−
+
−
+ · · · for −1 < x ≤ 1
ln(1 + x) =
k
2
3
4
k=1
Number ln 2
ln 2 =
∞
X
(−1)k+1
k=1
2.2
k
=1−
1 1 1
+ − + ···
2 3 4
Numerical Calculations
Outline
Contents
1 Taylor Polynomials
1.1 Taylor Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Remainder Term . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
3
2 Taylor Series
2.1 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Numerical Calculations . . . . . . . . . . . . . . . . . . . . . . .
4
4
6
6