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Calculus II Quiz
#7 April
22, 2005
Name_________________ R. Hammack Score
______
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(1) Decide if the following sequences
converge or diverge. In the case of convergence, state the limit if it can be
determined.
(a)
(n)
= (SEQUENCE
CONVERGES)
(b)
Odd terms approach ,
even terms approach .
THUS THE SEQUENCE DIVERGES
(c) (),(),(),(),(),...
The nth term is =.
=
0 (SEQUENCE CONVERGES)
(d)
,
,
,
,
...
Note: The series is defined recursively as =and =.
If the sequence has a limit L, then
L = =
which gives L = ,
or
= 2+L.
Thus

L  2 = 0
(L  2)(L + 1) = 0
So L is either 2 or 1.
But the series has positive terms, so it couldn't converge to a negative number.
Thus, if it converges, it must converge
to 2.
But does it converge?
From its definition, it should be clear that the sequence is increasing.
Also, notice that it has 2 is an upper bound, for:
=<
2
=<=
2
=<=
2
=<=
2, and so on.
Thus, it is an increasing sequence that is bounded above.
Therefore must converge, and as computed above, it converges to 2.