## 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.

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