MATH 1320 : Spring 2014
Lab 4
Lab Instructor : Kurt VanNess
Name:
Score:
Write all your solutions on a separate sheet of paper.
1. Suppose you know that a sequence {an } is an increasing sequence and all its terms lie between the
numbers 2 and 4. Explain why the sequence must have a limit? What can you say about the value of
the limit?
2. Find the limit of the sequence
(
3. Suppose that
∞
X
√
q
2,
√
2 2,
r q
)
√
2 2 2, . . . .
an (an 6= 0) is known to be a convergent series. Prove that
n=1
∞
X
1
is a divergent
a
n=1 n
series.
P
P
P
4. If
an and
bn are both divergent, is (an + bn ) necessarily divergent?
P∞
P∞
5. If the nth partial sum of a series n=1 an is sn = n−1
n=1 an .
n+1 , then find an and
6. Compute the sum of the series:
∞
X
2
. [Hint: Telescoping - Use partial fraction decomposition]
n(n + 2)
n=1
7. Determine whether the following series converge or not by using either the Integral Test or the Comparison Test:
∞
∞
X
X
1
n
√
,
.
3
5+n+1
n(ln
n)
n
n=2
n=1
8. (Alternating Series)
(a) Show that the series
∞
X
(−1)n
converges.
n2n
n=1
k
X
(−1)n
. Then S(k) is the partial sum of the series. Compute k so
n2n
n=1
that S(k) is within 0.01 of the sum of the series.
(b) Let S(k) be defined by:
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