## Bertrand Series 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:

Case 1: . Let . Hence, we have . Note that .

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.

Case 2: . Let . Hence, we have . Note that .

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.

Case 3: . Consider the function .

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:

Case 1: , then we have .

Since , then the series is not bounded,and therefore is divergent.

Case 2: , then we have ,

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.

Case 3: , we have ,

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 1.
If , the Bertrand series converges regardless of the value of or;
2.
If , the Bertrand series diverges regardless of the value of or;
3.
If , the Bertrand series converges if and only if .

For example, we have

1.
The series is divergent;

2.
The series is divergent;

3.
The series is convergent. [Trigonometry] [Calculus]
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Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard. Mohamed A. Khamsi
Tue Dec 3 17:39:00 MST 1996