42.
This is a geometric series which converges only for
)
ln
x
)
,
1, or
}
1
e
}
,
x
,
e
.
(a)
1
}
1
e
}
,
e
2
(b)
1
}
1
e
}
,
e
2
(c)
None
43.
n
5
13
3
10
9
?
365
?
24
?
3600
5
4.09968
3
10
17
ln (
n
1
1)
,
sum
,
1
1
ln
n
ln (4.09968
3
10
17
1
1)
,
sum
,
1
1
ln (4.09968
3
10
17
)
40.5548.
..
,
sum
,
41.5548.
..
40.554
,
sum
,
41.555
44.
Comparing areas in the figures, we have for all
n
$
1,
E
n
1
1
1
f
(
x
)
dx
,
a
1
1
…
1
a
n
,
a
1
1
E
n
1
f
(
x
)
dx
.
If the integral diverges, it must go to infinity, and the first
inequality forces the partial sums of the series to go to
infinity as well, so the series is divergent. If the integral
converges, then the second inequality puts an upper bound
on the partial sums of the series, and since they are a
nondecreasing sequence, they must converge to a finite sum
for the series. (See the explanation preceding Exercise 42 in
Section 9.4.)
45.
Comparing areas in the figures, we have for all
n
$
N
,
E
n
1
1
N
f
(
x
)
dx
,
a
N
1
…
1
a
n
,
a
N
1
E
n
N
f
(
x
)
dx
.
If the integral diverges, it must go to infinity, and the first
inequality forces the partial sums of the series to go to
infinity as well, so the series is divergent. If the integral
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 Spring '08
 GERMAN
 Geometric Series, Mathematical Series, lim, partial sums

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