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Calculus II Quiz
#8 April
29, 2005
Name_________________ R. Hammack Score
______
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Decide if the following series converge or diverge. In the case of convergence,
say whether the series converges conditionally or absolutely.
(1)
For k > 1, the series has positive terms.
Further, <
<
=
Therefore, the series converges by comparison with the convergent p-series
Since it converges and its terms are all positive, then it also converges
absolutely.
(2)
Using the ratio test =
=
=
=
0
Therefore the series converges. Since the terms are positive, it converges absolutely
(3)
This is an alternating series, with >
>
>...
and
=
=
0.
Therefore it converges by the alternating
series test.
However, |
|
=
=
+
+
+...
is the (divergent) harmonic series (minus the first term).
Therefore the original series converges conditionally.
(4)
-
+
-
+
-
+
...
Note that
does not exist, for odd terms approach 1 and even terms approach -1.
Therefore the series diverges by the divergence
test.