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49. (a)
When
t
5
1,
S
5
∑
‘
n
5
0
1
}
1
2
}
2
n
55
2.
(b)
S
converges when
)
}
1
1
t
t
}
)
,
1, or
)
t
)
,
)
1
1
t
)
.
For
t
,2
1, this inequality is equivalent to
2
t
(1
1
t
), which is always false.
For
2
1
#
t
,
0, the inequality is equivalent to
2
t
,
1
1
t
, which is true when
t
.2}
1
2
}
.
For
t
$
0, the inequality is equivalent to
t
,
1
1
t
,
which is always true.
Thus,
S
converges for all
t
1
2
}
.
(c)
For
t
1
2
}
, we have
S
5
∑
‘
n
5
0
1
}
1
1
t
t
}
2
n
}
(1
1
1
1
t
)
t
2
t
}5
1
1
t
,so
S
.
10 when
t
.
9.
50. (a)
Comparing
f
(
t
)
5 }
1
1
4
t
2
}
with
}
1
2
a
r
}
, the first term
is
a
5
4 and the common ratio is
r
52
t
2
.
First four terms: 4
2
4
t
2
1
4
t
4
2
4
t
6
General term: (
2
1)
n
(4
t
2
n
)
(b)
Note that
G
(0)
5
0, so the constant term of the power
series for
G
(
x
) will be 0. Integrate the terms for
f
(
x
) to
obtain the terms for
G
(
x
).
First four terms: 4
x
2 }
4
3
}
x
3
1 }
4
5
}
x
5
2 }
4
7
}
x
7
General term: (
2
1)
n
1
}
2
n
4
1
1
}
2
x
2
n
1
1
(c)
The series in part (a) converges when
)
2
t
2
)
,
1, so the
interval of convergence is (
2
1, 1).
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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

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