The Bertrand series are defined as
,
where and are real numbers. For example, the series
,
are Bertrand series. Here we will discuss the convergence and divergence of such series.
First, note that if , then the sequence is not bounded. Hence, it will not tend to 0, and therefore the series
,
is divergent. So, we will assume .
We have three cases:
.
Since , we get
.
So, for some , we have
,
which implies
.
By the p-test we know that the series is divergent (since p <1). The Basic Comparison Test implies that the series
is divergent.
.
Since , we get
.
So, for some , we have
,
which implies
.
By the p-test we know that the series is convergent (since p >1). The Basic Comparison Test implies that the series
is convergent.
.
It is easy to check that for large x (more precisely ), the function f(x) is decreasing. We will easily prove then that
,
and
,
where M is an integer such that f(x) is decreasing on .
Note that if , then we have
,
and if (we may take M=2), then we have
.
We have three cases:
.
Since , then the series is not bounded,and therefore is divergent.
,
but, since
,
for large n, we get
,
which means that the sequence of partial sums associated to the series
is bounded. Therefore, this series is convergent.
,
which implies
.
Since
,
we conclude that the partial sums associated to the series
are not bounded. Therefore, this series is divergent.
Let us summarize the above conclusions regarding the Bertrand series
For example, we have
is divergent;
is divergent;
is convergent.
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Mohamed A. Khamsi