Calculus Of One Real Variable By Pheng Kim Ving


Return To Contents
Go To Problems & Solutions
1. The Integral Test (IT) 
In Section
14.2 Harmonic Series, we established the divergence to infinity of the
harmonic series, the series of 1/n, by
comparing it to the improper integral of the function f(x) = 1/x for x > 0, which diverges to infinity. That's an
example of an
integral test, a test that establishes the behavior (convergence or divergence)
of a series by comparing it to an improper
integral whose behavior is known. The defining formula for the harmonic series
is a_{n} = 1/n, and that of the function is f(x) =
1/x, where x
is a positive real. The series and the function used in comparison of course
must have the same defining formula,
naturally except that the variable for the series is the positive integer n while that of the function is the positive real
x.

Fig. 1.1 If improper integral of f
converges, then series of a_{n}


Fig. 1.2 If improper integral of f
diverges to infinity, then series 
Note that when we want the sum of the areas of the
rectangles to be less than the area by f, we must of
course use the lower
rectangles, and that when we want the sum of the areas of the rectangles to be
greater than the area by f, we must of
course
use the upper rectangles. The test of the convergence or divergence of a series
done by comparing the series to the improper
integral of a function with the same defining formula is called the integral
test, abbreviated as IT.
Theorem 1.1 The
Integral Test (IT)

Proof
EOP
Example 1.1
Use the IT to establish:
Solution
EOS
The principal use of the IT is to establish the properties of the pseries, discussed below.
The Sum Of A Convergent Series And The Value of The Convergent Integral
^{{1.1}} Section 14.3 The Series Of 1/n^{2}.
Return To Top Of Page Go To Problems & Solutions
2. The pSeries 
^{{2.1}} Section 11.3 Theorem 2.1 i.
Corollary 2.1 The
pSeries

Proof
EOP
Example 2.1
Determine the convergence or divergence of the series:
Note
Solution
converges.
EOS
As this example illustrates, the LCT and the pseries property enable us to know at a glance
in advance the behavior of a
series of a rational function in n. The LCT
and the pseries property are used together
to determine the behavior of a series.
Example 2.2
Establish the convergence or divergence of this series:
Note
Solution
By the LCT, the series of 3/(n^{2} 100n) behaves like the convergent pseries of 1/n^{2} with p = 2 > 1, and thus converges. So
the given series converges.
EOS
Return To Top Of Page Go To Problems & Solutions
3. Remark 
The SCT and the LCT involve two series to be compared. The RooT
and the RaT involve one series and test its terms. The IT
involves one series and one improper integral to which the series is compared.
The principal use of the IT is to establish the
properties of the pseries.
Problems & Solutions 
1. Show that:
Solution
2. Show that:
Solution
Let u = ln x. Then du = (1/x) dx. Then:
3. Use the IT to determine whether the following series converges or
diverges:
Solution
4. Test the convergence of the series:
Solution
5. Determine the convergence or divergence of this series:
Solution
converges.
Return To Top Of Page Return To Contents