does the sequence necessarily become decreasing??

Consider a sequence of real numbers with the following properties:

1)

2)

3)

Define the sequence as .

Does necessarily become decreasing after a sufficiently large value of .

In other words, Does there exist a such that .

This is not a textbook question so i don't know the answer myself.

I can't think of an example of such that occurs *frequently*.

Re: does the sequence necessarily become decreasing??

Quote:

Originally Posted by

**abhishekkgp** Consider a sequence

of real numbers with the following properties:

1)

2)

3)

Define the sequence

as

.

Does

necessarily become decreasing after a sufficiently large value of

.

In other words, Does there exist a

such that

.

This is not a textbook question so i don't know the answer myself.

I can't think of an example of

such that

occurs

*frequently*.

The answer is No. In other words, such a sequence does not necessarily eventually decrease.

To start with, consider the sequence defined by for Then

and you can check by multiplying out the fractions that for all k. Thus there are infinitely many values of n for which

That example does not quite answer the original question, because the sequence is not strictly increasing (being constant throughout the interval ). However, in principle there is no difficulty in adjusting the sequence so as to make it strictly increasing. For each k, you can make very slightly smaller, without disturbing the property that . Then you can interpolate the values of linearly for so as to ensure that the sequence increases strictly.

Re: does the sequence necessarily become decreasing??

Quote:

Originally Posted by

**abhishekkgp** Consider a sequence

of real numbers with the following properties:

1)

2)

3)

Define the sequence

as

.

Does

necessarily become decreasing after a sufficiently large value of

.

In other words, Does there exist a

such that

.

This is not a textbook question so i don't know the answer myself.

I can't think of an example of

such that

occurs

*frequently*.

If we define then it must be...

a)

b) (1)

Now, if we define and in few steps we find that...

(2)

Because the series in (1) converges, it must be necesserly for some and and that means that so that is ...

Kind regards

Re: does the sequence necessarily become decreasing??

Quote:

Originally Posted by

**Opalg** The answer is No. In other words, such a sequence

does not necessarily eventually decrease.

To start with, consider the sequence

defined by

for

Then

and you can check by multiplying out the fractions that

for all k. Thus there are infinitely many values of n for which

That example does not quite answer the original question, because the sequence

is not strictly increasing (being constant throughout the interval

). However, in principle there is no difficulty in adjusting the sequence so as to make it strictly increasing. For each k, you can make

very slightly smaller, without disturbing the property that

. Then you can interpolate the values of

linearly for

so as to ensure that the sequence increases strictly.

that was brilliant!! thanks. it solved my question.

Re: does the sequence necessarily become decreasing??

Quote:

Originally Posted by

**chisigma** If we define

then it must be...

a)

b)

(1)

Now, if

we define

and in few steps we find that...

(2)

Because the series in (1) converges, it must be necesserly for some

and

and that means that

so that

is

...

Kind regards

thank you chisigma for your reply. It seems your answer to my question is "yes, the sequence necessarily becomes decreasing after a sufficiently large value of N." In that case i will take some time to review your solution. You may also want to read a very "close counterexample" provided by opalg.

thanks again.

Re: does the sequence necessarily become decreasing??

Re: does the sequence necessarily become decreasing??

Quote:

Originally Posted by

**chisigma** If we define

then it must be...

a)

b)

(1)

Now, if

we define

and in few steps we find that...

(2)

Because the series in (1) converges, it must be necesserly for some

and

and that means that

so that

is

...

May be it is necessary some more explanation of me so that I reported my post. The basic starting points are...

a)

b) (1)

... and the a) means that 'punctured infinite sums' like where...

(2)

... aren't adequate counterexamples...

Kind regards

Re: does the sequence necessarily become decreasing??

Quote:

Originally Posted by

**chisigma** May be it is necessary some more explanation of me so that I reported my post. The basic starting points are...

a)

b)

(1)

... and the a) means that 'punctured infinite sums' like

where...

(2)

... aren't adequate counterexamples...

If you want a more "adequate" counterexample, you could take

The fact is that the convergence of the series does **not** imply that there exist and such that for all .