The limit of its terms is By the test for divergence, the series
converges diverges
no conclusion
We determine if an infinite series diverges.
Definition of Infinite Series
Infinite Series We define If this limit is not finite or does not exist, then we say that
the infinite series diverges. If this limit is a finite number, then we say that the
infinite series converges.
The sum, , is called the partial sum of the infinite series . Thus, in words, we define
an infinite series to be the limit of its partial sums. This is reminiscent of an
improper integral which is also defined in terms of a limit.
Test for Divergence
In this section, we will learn a simple criterion for the divergence of an infinite series. The main idea is that in order for an infinite series to converge to a finite value, the terms in the series must approach zero. We now state this fact in its equivalent contrapositive form.
example 1 Consider the infinite series The degree of the numerator and denominator
of are equal (they are both one), so we can use the ratio of the lead coefficients to
determine that Since this limit is not zero, we can conclude that the series diverges
by the Test for Divergence.
(problem 1a) Consider the infinite series
(problem 1b) Consider the infinite series
The limit of its terms is By the test for divergence, the series
converges diverges
no conclusion
(problem 1c) Consider the infinite series
The limit of its terms is
By the test for divergence, the seriesconverges diverges no conclusion
(problem 1d) Consider the infinite series
The limit of its terms is By the test for divergence, the series
converges diverges
no conclusion
(problem 1e) Consider the infinite series
The limit of its terms is By the test for divergence, the series
converges diverges
no conclusion
example 2 Consider the infinite series The degree of the numerator (one) is less than
the degree of the denominator (two) so Since this limit is zero, the Test for
Divergence does not apply. The Test for Divergence is inconclusive in this
case.
(problem 2a) Consider the infinite series
The limit of its terms is By the test for divergence, the series
converges diverges
no conclusion
(problem 2b) Consider the infinite series
The limit of its terms is By the test for divergence, the series
converges diverges
no conclusion
(problem 2c) Consider the infinite series
The limit of its terms is By the test for divergence, the series
converges diverges
no conclusion
(problem 2d) Consider the infinite series
The limit of its terms is By the test for divergence, the series
converges diverges
no conclusion
Proof of the Test for Divergence
To prove the test for divergence, we will show that if converges, then the limit, , must equal zero. The logic is then that if this limit is not zero, the associated series cannot converge, and it therefore must diverge. We begin by considering the partial sums of the series, . Since the series converges, the limit of the partial sums exists and equals a finite value, which we will call . In mathematical symbols, where To proceed, we make the following two observations: and Putting these facts together, we can conclude that In the above limit, there is nothing special about the index (it was only helpful in dealing with the partial sums, ) and we can change it to to match statement in the theorem. We have therefore proved the theorem, since if then We will wrap up this section by examining the converse of the Test for Divergence. It was mentioned in the remark following the statement of the theorem above, that if the limit of the terms is zero, the associated series could either converge or diverge. We now present two classic series which exemplify this point. The two series are and In both cases, the limit of the terms is zero, but the first series converges while the second series diverges. We will explore both of these series in future sections (the first in the section on Geometric Series and the second in the section on the Integral Test).
Here is a detailed, lecture style video on the Test for Divergence:
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