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Math 153  Spring 2010
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Test 2A
21 April 2010
Answer the following questions. The answers must be clear, intelligible, and you must show your
work. Provide explanation for all your steps. Your grade will be determined by adherence to these
criteria. Use of books, notes and calculators is strictly forbidden. The point value of each problem
is given in the left hand margin.
(8 pts.)
1.
Determine whether the following series converges or diverges. State clearly what
test you are using and implement the test as clearly as you can. For full credit you must
show how you checked all the conditions required by the hypothesis of the convergence test
used.
1
X
n
=3
(
±
1)
n
ln
n
n
We use the Alternating series test.
Consider
f
(
x
) =
ln
x
x
; x
²
3
and compute
f
0
(
x
) =
1
x
³
x
±
ln
x
x
2
=
1
±
ln
x
x
2
Observe that 1
±
ln
x <
0 if
x > e
so the function is eventually decreasing. Therefore
b
n
+1
´
b
n
with
b
n
=
ln
n
n
. Next observe that
lim
n
!1
ln
n
n
= lim
n
!1
1
n
= 0
by l’Hopital rule.
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