James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

720 Chapter 11 Infinite Sequences and Series
So the sum of the areas of the rectangles is
1
1 2 1 1 2 2 1 1 3 2 1 1 4 2 1 1 5 2 1 ∙ ∙ ∙ − ò
n−1
If we exclude the first rectangle, the total area of the remaining rectangles is smaller
than the area under the curve y − 1yx 2 for x > 1, which is the value of the integral
y`
1 s1yx 2 d dx. In Section 7.8 we discovered that this improper integral is convergent and
has value 1. So the picture shows that all the partial sums are less than
1
1 1 y`
2 1
1
x 2 dx − 2
Thus the partial sums are bounded. We also know that the partial sums are increasing
(because all the terms are positive). Therefore the partial sums converge (by the Monotonic
Sequence Theorem) and so the series is convergent. The sum of the series (the
limit of the partial sums) is also less than 2:
ò
n−1
1
n 2
1
n 2 − 1 1 2 1 1 2 2 1 1 3 2 1 1 4 2 1 ∙ ∙ ∙ , 2
[The exact sum of this series was found by the Swiss mathematician Leonhard Euler
(1707–1783) to be 2 y6, but the proof of this fact is quite difficult. (See Problem 6 in the
Problems Plus following Chapter 15.)]
Now let’s look at the series
n s n − o n 1
i−1 si
5 3.2317
10 5.0210
50 12.7524
100 18.5896
500 43.2834
1000 61.8010
5000 139.9681
ò
n−1
1
− 1 1 1 1 1 1 1 1 1 1 ∙ ∙ ∙
sn s1 s2 s3 s4 s5
The table of values of s n suggests that the partial sums aren’t approaching a finite number,
so we suspect that the given series may be divergent. Again we use a picture for
confirmation. Figure 2 shows the curve y − 1ysx , but this time we use rectangles whose
tops lie above the curve.
y
y= 1
œ„x
FIGURE 2
0 1 2
3 4 5
area= 1 area=
1
area=
1
area=
1
œ„ 1 œ„ 2 œ„ 3 œ„ 4
x
The base of each rectangle is an interval of length 1. The height is equal to the value
of the function y − 1ysx at the left endpoint of the interval. So the sum of the areas of
all the rectangles is
1
1 1 1 1 1 1 1 1 1 ∙ ∙ ∙ − ò
s1 s2 s3 s4 s5 n−1
This total area is greater than the area under the curve y − 1ysx for x > 1, which is
1
sn
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