Bertrand Series

The Bertrand series are defined as

displaymath242,

where tex2html_wrap_inline244 and tex2html_wrap_inline246 are real numbers. For example, the series

displaymath248,

are Bertrand series. Here we will discuss the convergence and divergence of such series.

First, note that if tex2html_wrap_inline250 , then the sequence tex2html_wrap_inline252 is not bounded. Hence, it will not tend to 0, and therefore the series

displaymath242,

is divergent. So, we will assume tex2html_wrap_inline256 .

We have three cases:

Case 1: tex2html_wrap_inline258 . Let tex2html_wrap_inline260 . Hence, we have tex2html_wrap_inline262 . Note that

displaymath264.

Since tex2html_wrap_inline266 , we get

displaymath268.

So, for some tex2html_wrap_inline270 , we have

displaymath272,

which implies

displaymath274.

By the p-test we know that the series tex2html_wrap_inline276 is divergent (since p <1). The Basic Comparison Test implies that the series

displaymath242

is divergent.

Case 2: tex2html_wrap_inline282 . Let tex2html_wrap_inline260 . Hence, we have tex2html_wrap_inline286 . Note that

displaymath288.

Since tex2html_wrap_inline290 , we get

displaymath292.

So, for some tex2html_wrap_inline270 , we have

displaymath296,

which implies

displaymath298.

By the p-test we know that the series tex2html_wrap_inline276 is convergent (since p >1). The Basic Comparison Test implies that the series

displaymath242

is convergent.

Case 3: tex2html_wrap_inline306 . Consider the function

displaymath308.

It is easy to check that for large x (more precisely tex2html_wrap_inline312 ), the function f(x) is decreasing. We will easily prove then that

displaymath316,

and

displaymath318,

where M is an integer such that f(x) is decreasing on tex2html_wrap_inline324 .
Note that if tex2html_wrap_inline326 , then we have

displaymath328,

and if tex2html_wrap_inline330 (we may take M=2), then we have

displaymath334.

We have three cases:

Case 1: tex2html_wrap_inline336 , then we have

displaymath338.

Since tex2html_wrap_inline340 , then the series tex2html_wrap_inline342 is not bounded,and therefore is divergent.

Case 2: tex2html_wrap_inline344 , then we have

displaymath346,

but, since

displaymath348,

for large n, we get

displaymath352,

which means that the sequence of partial sums associated to the series

displaymath354

is bounded. Therefore, this series is convergent.

Case 3: tex2html_wrap_inline330 , we have

displaymath358,

which implies

displaymath360.

Since

displaymath362,

we conclude that the partial sums associated to the series

displaymath364

are not bounded. Therefore, this series is divergent.

Let us summarize the above conclusions regarding the Bertrand series

displaymath366

1.
If tex2html_wrap_inline368 , the Bertrand series converges regardless of the value of tex2html_wrap_inline370 or;
2.
If tex2html_wrap_inline372 , the Bertrand series diverges regardless of the value of tex2html_wrap_inline370 or;
3.
If tex2html_wrap_inline376 , the Bertrand series converges if and only if tex2html_wrap_inline378 .

For example, we have

1.
The series

displaymath364

is divergent;

2.
The series

displaymath382

is divergent;

3.
The series

displaymath384

is convergent.

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Mohamed A. Khamsi
Tue Dec 3 17:39:00 MST 1996

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