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Calculus 1 Problems & Solutions – Chapter 9 – Section 9.5

9.5
The Integral Test

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Review

1. The Integral Test (IT)

In Section 9.2 Heading 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
also converges.

Fig. 1.2

If improper integral of f diverges to infinity, then series
of a_n 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 9.3 The Series Of 1/n².

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2. The p-Series

^{2.1} Section 6.7.2 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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