Fall 2010 Math 152
2
Week in Review XI
Section 10.7
Key Concepts:
courtesy: David J. Manuel
1. The Taylor Series (power series representation) of any differentiable function f at x = a
∞
X
f (n) (a)
(x − a)n .
is given by
n!
n=0
(covering 10.7, 10.9)
1
Section 10.6
2. If a = 0 above, the series is called the Maclaurin Series of f .
3. The N th partial sum of the Taylor Series is
called the N th degree Taylor Polynomial of
f . The remainder Rn (x) = f (x) − Tn (x)
(NOTE: If |Rn (x)| → 0 then the Taylor Series
converges to f (x) for all |x − a| < R)
Key Concepts:
X
1. If
cn (x − a)n
converges to f (x)
with
radius
of
convergence
R, then
X
n−1
cn n(x − a)
converges to f ′ (x) with
radius of convergence R.
X
2. If
cn (x − a)n
converges to f (x)
with radius of convergence R,
then
ˆ
X cn
n+1
(x − a)
converges to
f (x) dx
n+1
with radius of convergence R.
4. If f (n+1) (x)| ≤ M for |x − a| < R, then
M
|Rn (x)| ≤
|x − a|n+1 for |x − a| < R.
(n + 1)!
Examples:
1. Find the Taylor Series for the given functions
at the given centers, and determine the radius
of convergence:
3. KEY to creating many power series: Geomet∞
X
a
.
ric series formula
arn−1 =
1−r
n=1
(a) f (x) = ex , x = 1
(b) f (x) = ln x, x = 1
´x
2
(c) f (x) = 0 e−t dt
Examples:
(d) f (x) = sin2 x (HINT: use a helpful identity)
1. Find a power series for the following functions
and determine the radius and interval of convergence:
2. Prove the Taylor Series in (c) converges to f
by showing |Rn (x)| → 0
(a) f (x) = arctan x
1
(b) f (x) =
(2 − x)2
3
(c) f (x) = ln(1 + x2 )
2.
Section 10.9
Key Concepts:
(a) Use
ˆ 1
a power series to compute
1
dx.
4
0 1+x
(b) How many terms of the series are needed
to approximate the integral to within
.01?
1. The N th degree Taylor Polynomial of
f at x = a is given by TN (x) =
N
X
f (n) (a)
(x − a)n (Basically, it is the N th
n!
n=0
partial sum of the Taylor Series!)
1
2. If RN (x) = |f (x) − TN (x)|, we can estimate
RN (x) on a given interval one of three ways:
(a) Graphical (Matlab)
(b) If Alternating, use |s − sN | ≤ |aN +1 |
(c) Otherwise,
use Taylor’s Inequality
M
|x − a|n+1
|Rn (x)| ≤
(n + 1)!
Examples:
1. Find the fourth-degree Taylor Polynomial of
1
f (x) =
centered at x = 0.
4−x
2. Find the
√ third-degree Taylor Polynomial of
f (x) = x centered at x = 1.
3. ˆ
Use
1
a
e
Maclaurin
−x2 /2
Series
to
determine
dx correct to within 0.001.
0
4. Find the fourth-degree
Taylor
Polynomial of
ex + e−x
f (x) = cosh x =
at x = 0 and
2
determine an upper bound on RN (x) on the
interval x ∈ (0, 1).
2