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19.
Converges absolutely, since
∑
‘
n
5
1
n
2
1
}
2
3
}
2
n
converges by the
Ratio Test:
lim
n
→
‘
)
}
a
a
n
1
n
1
}
)
5
(
n
1
1)
2
1
}
2
3
}
2
n
1
1
?
}
n
1
2
}
1
}
3
2
}
2
n
5 }
2
3
}
,
1.
20.
Converges conditionally.
If
u
n
5 }
n
l
1
n
n
}
, then {
u
n
} is a decreasing sequence of
positive terms with lim
n
→
‘
u
n
5
0, so
∑
‘
n
5
2
(
2
1)
n
1
1
}
n
l
1
n
n
}
converges by the Alternating Series Test.
But
∑
‘
n
5
2
}
n
l
1
n
n
}
diverges by the integral test, since
E
‘
2
}
x
l
1
n
x
}
dx
5
lim
b
→
‘
3
ln
)
ln
x
)
4
5‘
.
21.
Diverges by the
n
thTerm Test, since lim
n
→
‘
}
2
n
n
!
}5‘
and so the
terms do not approach 0.
22.
Converges absolutely, since
∑
‘
n
5
1
)
}
si
n
n
2
n
}
)
converges by direct
comparison to
∑
‘
n
5
1
}
n
1
2
}
, which converges as a
p
series with
p
5
2.
23.
Converges conditionally:
If
u
n
5
}
1
1
1
ˇ
n
w
}
, then {
u
n
} is a decreasing sequence of
positive terms with lim
n
→
‘
u
n
5
0, so
∑
‘
n
5
1
}
1
(
1
2
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This note was uploaded on 10/05/2011 for the course MAC 1147 taught by Professor German during the Spring '08 term at University of Florida.
 Spring '08
 GERMAN
 Geometric Series

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