SECTION 8.3
THE INTEGRAL AND
COMPARISON TESTS
INFINITE SEQUENCES AND SERIES
In general, it is difficult to find the exact sum of
a series.



We were able to accomplish this for geometric
series and the series ∑1/[n(n+1)].
This is because, in each of these cases, we can find
a simple formula for the nth partial sum sn.
sn
Nevertheless, usually, it isn’t easy to compute lim
n 
8.3 P2
INFINITE SEQUENCES AND SERIES
Therefore, in this section and the next we
develop tests that enable us to determine
whether a series is convergent or divergent
without explicitly finding its sum.
In this section we deal only with series with
positive terms, so the partial sums are increasing.
In view of the Monotonic Sequence Theorem, to
decide whether a series is convergent or
divergent, we need to determine whether the
partial sums are bounded or not.
8.3 P3
TESTING WITH AN INTEGRAL
Let’s investigate the series whose terms are the
reciprocals of the squares of the positive
integers:

1 1 1 1 1 1
 2  2  2  2  2  ...

2
1 2 3 4 5
n 1 n

There’s no simple formula for the sum sn of the first
n terms.
8.3 P4
INTEGRAL TEST
However, the computer-generated values given
here suggest that the partial sums are
approaching near 1.64 as n → ∞.


So, it looks as if the series is
convergent.
We can confirm this impression
with a geometric argument.
8.3 P5
INTEGRAL TEST
We can confirm this impression with a
geometric argument.
Figure 1 shows the curve y = 1/x2 and
rectangles that lie below the curve.
8.3 P6
INTEGRAL TEST
The base of each rectangle is an interval of
length 1.
The height is equal to the value of the function y
= 1/x2 at the right endpoint of the interval.
8.3 P7
INTEGRAL TEST
Thus, the sum of the areas of the rectangles is:

1 1 1 1 1
1
 2  2  2  2  ...   2
2
1 2 3 4 5
n 1 n
8.3 P8
INTEGRAL TEST
If we exclude the first rectangle, the total area of
the remaining rectangles is smaller than the area
under the curve y = 1/x2 for x ≥ 1, which is the
value of the integral


1

(1/ x 2 ) dx
In Section 6.6, we discovered that this improper
integral is convergent and has value 1.
8.3 P9
INTEGRAL TEST
So, the image shows that all the partial sums are
less than
 1
1
  2 dx  2
2
1 x
1

Therefore, the partial sums are bounded.
We also know that the partial sums are
increasing (as all the terms are positive).
Thus, the partial sums converge (by the
Monotonic Sequence Theorem).

So, the series is convergent.
8.3 P10
INTEGRAL TEST
The sum of the series (the limit of the partial
sums) is also less than 2:

1 1 1 1 1
 2  2  2  2  ...  2

2
1 2 3 4
n 1 n
8.3 P11
INTEGRAL TEST
The exact sum of this series was found by the
mathematician Leonhard Euler (1707–1783) to
be p2/6.

However, the proof of this fact is beyond the scope
of this book.
8.3 P12
INTEGRAL TEST
Now, let’s look at this series:

1
1
1
1
1
1
 



 ...

n
1
2
3
4
5
n 1
8.3 P13
INTEGRAL TEST
The table of values of sn 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.
8.3 P14
INTEGRAL TEST
Figure 2 shows the curve y = 1/ x .
However, this time, we use rectangles whose
tops lie above the curve.
8.3 P15
INTEGRAL TEST
The base of each rectangle is an interval of
length 1.
The height is equal to the value of the function
y = 1/ x at the left endpoint of the interval.
8.3 P16
INTEGRAL TEST
So, the sum of the areas of all the rectangles is:

1
1
1
1
1
1




 ...  
1
2
3
4
5
n
n 1
8.3 P17
INTEGRAL TEST
This total area is greater than the area under the
curve y = 1/ x for x ≥ 1, which is equal to the

integral  (1/ x ) dx
1


However, we know from Section 6.6 that this
improper integral is divergent.
In other words, the area under the curve is infinite.
8.3 P18
INTEGRAL TEST
Thus, the sum of the series must be infinite.

That is, the series is divergent.
The same sort of geometric reasoning that we
used for these two series can be used to prove
the following test.

The proof is given at the end of the section.
8.3 P19
THE INTEGRAL TEST
Suppose f is a continuous, positive, decreasing
function
on [1, ∞) and let an = f(n). Then, the series

an is convergent if and only if the improper


n 1
integral  f ( x) dx is convergent. In other words,
1


(a) If 1 f ( x) dx is convergent, then  anis
n 1
convergent.


(b) If 1 f ( x) dx is divergent, then  an is
n 1
divergent.
8.3 P20
NOTE
When we use the Integral Test, it is not
necessary to start the series or the integral at n =
1.

For instance, in testing the series

1
we use

2
n  4 ( n  3)


4
1
dx
2
( x  3)
8.3 P21
NOTE
Also, it is not necessary that f be always
decreasing.



What is important is that f be ultimately decreasing,
that is, decreasing for x larger than some number N.

Then,  an is convergent.
n N

So,  an is convergent by Note 4 of Section 8.2
n 1
8.3 P22