Summary for §8.5 - §8.8
Math 1321 (Qinghai Zhang)
(m)
Lemma 5. ∀m = 0, 1, 2, . . . , n, Rn (a) = 0.
This week we applied the idea of “approximation
with increasing accuracies” to functions.
1
2013-JAN-18
Theorem 6. Let Tn be the nth Taylor polynomial for
f (x) at a.
Power series
lim Rn (x) = 0 ⇔ f (x) = lim Tn (x).
n→∞
n→∞
(8)
Definition 1. A power series centered at a is a series of Theorem 7 (Rolle’s). If a function f : R → R satisfies
the form
∞
X
(i) f is continuous on [a, b] and differentiable on (a, b),
p(x) =
cn (x − a)n ,
(1)
(ii) f (a) = f (b),
n=0
where cn ’s are called the coefficients. The interval of then ∃x ∈ (a, b) s.t. f 0 (x) = 0.
convergence is the set of x values for which the series
Theorem 8 (Taylor’s theorem - Lagrangian form). Conconverges:
sider a function f : R → R. If f (n+1) (x) exists on the
Ic (f ) = {x | f (x) converges}.
(2) interval I = (a − d, a + d), and f (n) (x) is continuous on
Theorem 2 (interval of convergence). There are only [a − d, a + d], then ∀x ∈ I, ∃y ∈ I s.t.
three possibilities
P∞ for the interval of convergence of a
f (n+1) (y)
power series n=0 cn (x − a)n :
(x − a)n+1 .
(9)
Rn (x) =
(n + 1)!
(i) Ic = {a},
Proof. For any fixed x 6= a, ∃M s.t.
(ii) Ic = (−∞, +∞),
M (x − a)n+1
Rn (x) = f (x) − Tn (x) =
.
(10)
(iii) Ic = (a − R, a + R) or [a − R, a + R) or (a − R, a + R]
(n + 1)!
or [a − R, a + R].
Consider function
R is called the Radius of convergence. R = 0, +∞ for
M (t − a)n+1
cases (i) & (ii), respectively.
g(t) = Rn (t) −
.
(n + 1)!
Theorem 3 (Term-by-term
differentiation
and
integraP∞
tion). If f (x) = n=0 cn (x − a)n has radius of conver- By Lemma 5, g(a) = 0 and g (k) (a) = 0 for k = 0, 1, . . . , n.
gence R > 0, then f (x) is differentiable on (a − R, a + R). Also, g(x) = 0 by (10). By Rolle’s theorem,
∞
X
∃x1 ∈ (a, x) s.t. g 0 (x1 ) = 0.
ncn (x − a)n−1 ,
(3)
f 0 (x) =
n=0
If x < a, change (a, x) above to (x, a). Similarly,
Z
∞
X
(x − a)n+1
∃x2 ∈ (a, x1 ) s.t. g (2) (x2 ) = 0.
.
(4)
f (x)dx = C +
cn
n
+
1
n=0
Repeatedly using Rolle’s theorem,
Furthermore,
the
radii
of
convergence
of both f 0 (x) and
R
∃xn+1 ∈ (a, xn ) s.t. g (n+1) (xn+1 ) = 0.
(11)
f (x)dx are R.
(n+1)
2
Since Tn
(t) = 0, g (n+1) (t) = M − f (n+1) (t). (11)
completes the proof by showing the existence of y with
y = xn+1 .
Taylor Series
Definition 4. Consider a function f : (a−R, a+R) → R
with R > 0. If f (n) (x) exists at x = a, then
Tn (x) =
n
X
f (k) (a)
k=0
k!
(x − a)k
The proof is optional, but I hope you find it elegant.
Theorem 9 (Taylor’s inequality). If ∃M < ∞ s.t. ∀x ∈
(5) [a − d, a + d], f (n+1) (x) ≤ M , then ∀x ∈ [a − d, a + d],
|Rn (x)| ≤
is called the nth Taylor polynomial for f (x) at a.
∞
X
f (k) (a)
M
|x − a|n+1 .
(n + 1)!
(12)
(6) Consequently f (x) = limn→∞ Tn .
k=0
Theorem 10 (The Binomial Series). ∀k ∈ R, |x| < 1,
is called the Taylor series for f (x) at a. The remainder
∞ ∞ Qn−1
X
X
k n
k
i=0 (k − i) n
of Tn in approximating f (x) is
(1 + x) =
x =1+
x (13)
n
n!
n=0
n=1
Rn (x) = f (x) − Tn (x).
(7)
lim Tn (x) =
n→∞
k!
(x − a)k
1