Fall 2010 Math 152
2. The hundredth partial sum of 1c above is
s100 ≈ .049787. Estimate how close this is
to the actual sum of the series.
Week in Review X
3. How many terms of the series in 1a above
are needed to approximate the exact sum to
within .001?
courtesy: David J. Manuel
(covering 10.4, 10.5, 10.6)
1
2
Section 10.4
Section 10.5
Key Concepts:
Key Concepts:
1. Power
Series:
X
cn (x − a)n
1. Alternating Series
X Test: If an decreases
and an → 0, then
(−1)n an is convergent
2. Alternating
X
If
(−1)n an
sN
=
N
X
Series
converges
(−1)n an (an
2. Interval of Convergence:
the set of all valX
ues of x for which
cn (x − a)n is convergent. Possibilities:
Estimation:
to s and
>
0),
A series of the form
(a) only when x = a
then
(b) x ∈ (−∞, ∞)
n=1
|s − sN | ≤ aN +1
(c) x ∈ (a − r, a + r) for some positive r
(possibly including endpoints)
X
3. Absolute Convergence: If
|an | is conX
3. Radius of Convergence: The largest value
an is absolutely convergent
vergent, then
of
Xr such that, if x ∈ (a − r, a + r), then
(and hence, convergent)
cn (x − a)n is convergent. (Case (a): r =
an+1 0, Case (b): r = ∞)
. If L > 1,
4. Ratio Test: Let L = lim n→∞
a
n
X
X
an is divergent; if L < 1,
an is abso- Examples:
lutely convergent.
1. Find the radius and interval of convergence of
each of the following:
Examples:
1. Determine if the following series
lutely convergent, convergent (but
lutely convergent), or divergent.
test used and show all conditions
fied.
(a)
∞
X
are absonot absoState the
are satis-
n+1
(−1)
2n + 1
n=1
∞
X
22n
(b)
n3n+1
n=1
(a)
∞
X
(x + 2)n
2n
n=0
(b)
∞
X
(−1)n x2n+1
2n + 1
n=0
(c)
∞
X
3n (x − 1)n
n ln n
n=2
(d)
∞
X
2n (x − 3)n
n!
n=0
X
2. Given
cn (x − 2)n is convergent when x =
5 and divergent when x = −4, determine if
the following are convergent, if possible:
∞
X
(−3)n
(c)
n!
n=0
1
3
(a)
X
cn
(b)
X
cn 9 n
(c)
X
cn (−2)n
Section 10.6
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.
3. KEY to creating many power series: Geomet∞
X
a
ric series formula
arn−1 =
.
1
−
r
n=1
Examples:
1. Find a power series for the following functions
and determine the radius and interval of convergence:
(a) f (x) = arctan x
1
(b) f (x) =
(2 − x)2
(c) f (x) = ln(1 + x2 )
2.
(a) Use
ˆ 1
a power series to compute
1
dx.
1
+
x4
0
(b) How many terms of the series are needed
to approximate the integral to within
.01?
2