Consider the series and its associated sequence of partial sums . Here we will assume that the numbers we are about to add are positive, that is, for any . It is clear that the process of generating the partial sums will lead to an increasing sequence, that is,
,
for any . Our previous knowledge about increasing sequences
implies the following fundamental result:
,
for any .
This result has many implications. For example, we have the following
result (called The Basic Comparison Test):
,
then the series is convergent.
,
then the series is divergent.
Recall that in previous pages, we showed the following
,
which means that the series is divergent.
Example: Show that the series
is divergent.
Answer: We have , for any ; hence
.
Since is divergent, we
deduce from the Basic Comparison Test, that is divergent.
Example: Show that the series
is convergent.
Answer: Since
and the geometric series
is convergent, then the series is convergent (using the Basic Comparison
Test).
The next result (known as The p-Test) is as fundamental as the
previous ones. Usually we combine it with the previous ones or new
ones to get the desired conclusion.
Consider the positive series (called the p-series) . Since the limit of the numbers must add to 0, in order to expect convergence, we assume that p > 0. The next result deals with convergence or divergence of the series
when p >0.
The proof of the above result is very instructive by itself. So let
us discuss how this works:
.
It is easy to check that f(x) is decreasing on . Hence, for any , we have for any ,
which implies
,
that is,
.
Using this inequality, we get
and
.
If , then we have
and if p=1, then we have
.
.
Since , then the series is not bounded,and therefore it is divergent.
but, since
,
we get
,
which means that the sequence of partial sums associated to the series is bounded. Therefore, the series is convergent.
,
which shows that the sequence of partial sums is not bounded.
Example: Discuss the convergence or divergence of
.
Answer: It is not hard to show that for any , we have . Then, we have
.
Since, by the p-Test, the series is
convergent, the Basic comparison Test implies that is convergent.
The last result on positive series may be the most useful of all.
Indeed, the Limit Test should be always in mind when it comes to
cleaning up some undesirable terms.
Before we state this test, we need a new notation. Indeed, we will
say that the two sequences and are equivalent, or , if and only if,
.
.
Then converges, if and only if, converges.
Example: Determine whether the series
is convergent or not.
Answer: Note that when n is large we have and . Then it is easy to check that
.
Using the p-test we get that the series is convergent. Hence, by the Limit-test, we deduce
the convergence of the series .
Example: Determine whether the series
is convergent or not.
Answer: We know that
.
This limit means that when x is very small , then (a very useful conclusion in physics, for example, when dealing the motion of the pendulum). So when n is large, 1/n will be small and therefore . This clearly implies that
Since the series is
divergent, the limit-test implies that the series
is
divergent.
Remark: It should be appreciated that, without the Limit-test, it would be very hard to check the convergence.
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Mohamed A. Khamsi