Calculus Of One Real Variable – By Pheng Kim Ving
Chapter 14: Infinite Series – Section 14.5: The Integral Test


14.5
The Integral Test

 

 

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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 an = 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 an
also converges.

 

Fig. 1.2

 

If improper integral of f diverges to infinity, then series
of an also diverges to infinity.

 

 

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 p-series, discussed below.

 

The Sum Of A Convergent Series And The Value of The Convergent Integral

 

 

{1.1} Section 14.3 The Series Of 1/n2.

 

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2. The p-Series

 

 

{2.1} Section 11.3 Theorem 2.1 i.

 

Corollary 2.1 – The p-Series

 

 

 

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 p-series 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 p-series 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/(n2 – 100n) behaves like the convergent p-series of 1/n2 with p = 2 > 1, and thus converges. So
the given series converges.
EOS

 

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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 p-series.

 

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Problems & Solutions

 

1. Show that:

 

 

Solution

 

 

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2. Show that:

 

    

 

Solution

 

 

Let u = ln x. Then du = (1/x) dx. Then:

 

 

 

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3. Use the IT to determine whether the following series converges or diverges:

 

     

 

Solution

 

 

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4. Test the convergence of the series:

 

   

 

Solution

 

 

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5. Determine the convergence or divergence of this series:

 

   

 

Solution

 

 

 

converges.

 

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