Fall 2010 Math 152
4. Compute the sum
∞ n
X
1
n=0
2
n 2
or
+
3
show the series diverges.
Week in Review VIII
5. Write the repeating decimal 0.51 as a fraction.
courtesy: David J. Manuel
(covering 10.2, 10.3)
2
1
Section 10.3
Section 10.2
Key Concepts:
Key Concepts:
1. Infinite Series:
∞
X
an . Define sN =
n=1
The series converges to s if and only
s.
N
X
n=1
if sN
1. Integral Test: If an = f (n), where f isX
continuous, positive, and decreasing, then
an
ˆ ∞
is convergent if and only if
f (x) dx con1
verges
an .
→
2. Remainder
for
Integral
Test:
X
If
an
converges
to
s,
then
ˆ ∞
ˆ ∞
f (x) dx ≤ s − sN ≤
f (x) dx.
2. Series we can sum:
(a) Geometric:
∞
X
arn =
n=1
if |r| < 1.
(b) Telescoping:
∞
X
a
if and only
1−r
N +1
(bn+1 − bn )
4. Comparison Test: if 0 ≤ an ≤ bn , then:
X
X
(a) If
bn is convergent, then
an is
convergent
X
X
(b) If
an is divergent, then
bn is divergent
n=1
3. Test for Divergence: If
then an → 0.
X
N
X 1
3. P-Test:
is convergent if and only if
np
p > 1.
an is convergent,
Examples:
5. Limit Comparison Test: If anX
> 0 and bX
n > 0
an
and 0 < lim
< ∞, then
an and
bn
n→∞ bn
either both converge or both diverge.
∞
X
3
1. Given the series
, find a formula
2+n
n
n=1
for sN and determine if the series converges
or diverges. If it converges, find the sum.
2. Given a series whose partial sums are given by Examples:
sN = ln(N + 1), find an and determine if the
series converges or diverges. If it converges,
1. Determine whether the following series confind the sum.
verge or diverge. State the name of the test
used and show all conditions are met.
3. A student begins running laps on a 440 yard
track. He finishes the first lap in 1 minute,
but finds that for each subsequent minute he
9
the distance of the previous
only covers
10
minute. If the student could run indefinitely,
what distance will he cover?
1
∞
X
(a)
1
1
+
n2
n=0
(b)
∞
X
1
1/n
2
n=1
(c)
(d)
(e)
(f)
∞
X
n=1
∞
X
2n
1
+n
1
√
3
n
+2
n=1
∞
X
n=1
∞
X
n=1
n2
2n
√
− n+1
sin
π
n
2. The partial sum s10 of
∞
X
2
ne−n is 0.40488.
n=1
Find an upper and lower bound on the sum
of the series.
3. How many terms of the series are needed to
∞
X
1
approximate
to within 2 × 10−9 ?
5
n
n=1
2