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58. continued
(c)
f
1
2}
1
4
}
2
5 }
2
3
}
, so one percent is approximately 0.0067. It
takes 7 terms (up through degree 6). This can be found
by trial and error. Also, for
x
52}
1
4
}
, the series is the
alternating series
∑
‘
n
5
0
1
2}
1
2
}
2
n
. If you use the Alternating
Series Estimation Theorem, it shows that 8 terms (up
through degree 7) are sufficient since
)
2}
1
2
}
)
8
,
0.0067.
It is also a geometric series, and you could use the
remainder formula for a geometric series to determine
the number of terms needed. (See Example 2 in
Section 9.3.)
59. (a)
lim
n
→
‘
)
}
a
a
n
1
n
1
}
)
5
lim
n
→
‘
?
}
)
x
n
)
n
!
n
n
}
5
lim
n
→
‘
5
)
x
)
lim
n
→
‘
1
}
n
1
n
1
}
2
n
5
)
x
)
e
The series converges for
)
x
)
e
,
1, or
)
x
)
, }
1
e
}
, so the
radius of convergence is
}
1
e
}
.
(b)
f
1
2}
1
3
}
2
<
2}
1
3
}
?
}
1
1
}
1
1
2}
1
3
}
2
2
?
}
2
2
2
!
} 1
1
2}
1
3
}
2
3
?
}
3
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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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