# does the sequence necessarily become decreasing??

• Dec 28th 2011, 10:08 AM
abhishekkgp
does the sequence necessarily become decreasing??
Consider a sequence $\displaystyle (x_n)$ of real numbers with the following properties:
1)$\displaystyle x_n>0 \, \forall \, n$
2)$\displaystyle x_n<1 \, \forall \, n$
3)$\displaystyle x_{n+1}>x_n \, \forall \, n$

Define the sequence $\displaystyle (y_n)$ as $\displaystyle y_n=\frac{x_n}{n}$.

Does $\displaystyle (y_n)$ necessarily become decreasing after a sufficiently large value of $\displaystyle n$.

In other words, Does there exist a $\displaystyle N$ such that $\displaystyle y_{n+1} \leq y_n \, \forall \, n \, \geq N$.

This is not a textbook question so i don't know the answer myself.
I can't think of an example of $\displaystyle (x_n)$ such that $\displaystyle y_{n+1} > y_n$ occurs frequently.
• Dec 28th 2011, 11:14 AM
Opalg
Re: does the sequence necessarily become decreasing??
Quote:

Originally Posted by abhishekkgp
Consider a sequence $\displaystyle (x_n)$ of real numbers with the following properties:
1)$\displaystyle x_n>0 \, \forall \, n$
2)$\displaystyle x_n<1 \, \forall \, n$
3)$\displaystyle x_{n+1}>x_n \, \forall \, n$

Define the sequence $\displaystyle (y_n)$ as $\displaystyle y_n=\frac{x_n}{n}$.

Does $\displaystyle (y_n)$ necessarily become decreasing after a sufficiently large value of $\displaystyle n$.

In other words, Does there exist a $\displaystyle N$ such that $\displaystyle y_{n+1} \leq y_n \, \forall \, n \, \geq N$.

This is not a textbook question so i don't know the answer myself.
I can't think of an example of $\displaystyle (x_n)$ such that $\displaystyle y_{n+1} > y_n$ occurs frequently.

The answer is No. In other words, such a sequence $\displaystyle (y_n)$ does not necessarily eventually decrease.

To start with, consider the sequence $\displaystyle (x_n)$ defined by $\displaystyle x_n = \tfrac k{k+1}$ for $\displaystyle k^2+1\leqslant n\leqslant (k+1)^2\ (k=0,1,2,\ldots).$ Then

$\displaystyle y_{k^2+1} = \frac{x_{k^2+1}}{k^2+1} = \frac k{(k+1)(k^2+1)},\qquad y_{k^2} = \frac{x_{k^2}}{k^2} = \frac {k-1}{k^3},$

and you can check by multiplying out the fractions that $\displaystyle y_{k^2+1} > y_{k^2}$ for all k. Thus there are infinitely many values of n for which $\displaystyle y_{n+1}>y_n.$

That example does not quite answer the original question, because the sequence $\displaystyle (x_n)$ is not strictly increasing (being constant throughout the interval $\displaystyle k^2+1\leqslant n\leqslant (k+1)^2$). However, in principle there is no difficulty in adjusting the sequence so as to make it strictly increasing. For each k, you can make $\displaystyle x_{k^2+1}$ very slightly smaller, without disturbing the property that $\displaystyle y_{k^2+1} > y_{k^2}$. Then you can interpolate the values of $\displaystyle x_n$ linearly for $\displaystyle k^2+1\leqslant n\leqslant (k+1)^2$ so as to ensure that the sequence increases strictly.
• Dec 28th 2011, 06:47 PM
chisigma
Re: does the sequence necessarily become decreasing??
Quote:

Originally Posted by abhishekkgp
Consider a sequence $\displaystyle (x_n)$ of real numbers with the following properties:
1)$\displaystyle x_n>0 \, \forall \, n$
2)$\displaystyle x_n<1 \, \forall \, n$
3)$\displaystyle x_{n+1}>x_n \, \forall \, n$

Define the sequence $\displaystyle (y_n)$ as $\displaystyle y_n=\frac{x_n}{n}$.

Does $\displaystyle (y_n)$ necessarily become decreasing after a sufficiently large value of $\displaystyle n$.

In other words, Does there exist a $\displaystyle N$ such that $\displaystyle y_{n+1} \leq y_n \, \forall \, n \, \geq N$.

This is not a textbook question so i don't know the answer myself.
I can't think of an example of $\displaystyle (x_n)$ such that $\displaystyle y_{n+1} > y_n$ occurs frequently.

If we define $\displaystyle \delta_{x}(n)= x_{n+1}-x_{n}$ then it must be...

a) $\displaystyle \delta_{x}(n)>0\ \forall n$

b) $\displaystyle \lim_{n \rightarrow \infty} x_{n}= x_{1} + \sum_{k=1}^{\infty} \delta_{x}(k)= x_{0}\le 1$ (1)

Now, if $\displaystyle y_{n}=\frac{x_{n}}{n}$ we define $\displaystyle \delta_{y}(n)= y_{n+1}-y_{n}$ and in few steps we find that...

$\displaystyle \delta_{y}(n)= \frac{\delta_{x}(n)-y_{n}}{n+1}$ (2)

Because the series in (1) converges, it must be necesserly for some $\displaystyle \varepsilon>0$ and $\displaystyle N$ $\displaystyle \delta_{x}(n)<\frac{\varepsilon}{n}\ \forall n>N$ and that means that $\displaystyle \delta_{x}(n)<y_{n}\ \forall n>N$ so that $\displaystyle \forall n>N$ is $\displaystyle \delta_{y}(n)<0$...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
• Dec 28th 2011, 11:54 PM
abhishekkgp
Re: does the sequence necessarily become decreasing??
Quote:

Originally Posted by Opalg
The answer is No. In other words, such a sequence $\displaystyle (y_n)$ does not necessarily eventually decrease.

To start with, consider the sequence $\displaystyle (x_n)$ defined by $\displaystyle x_n = \tfrac k{k+1}$ for $\displaystyle k^2+1\leqslant n\leqslant (k+1)^2\ (k=0,1,2,\ldots).$ Then

$\displaystyle y_{k^2+1} = \frac{x_{k^2+1}}{k^2+1} = \frac k{(k+1)(k^2+1)},\qquad y_{k^2} = \frac{x_{k^2}}{k^2} = \frac {k-1}{k^3},$

and you can check by multiplying out the fractions that $\displaystyle y_{k^2+1} > y_{k^2}$ for all k. Thus there are infinitely many values of n for which $\displaystyle y_{n+1}>y_n.$

That example does not quite answer the original question, because the sequence $\displaystyle (x_n)$ is not strictly increasing (being constant throughout the interval $\displaystyle k^2+1\leqslant n\leqslant (k+1)^2$). However, in principle there is no difficulty in adjusting the sequence so as to make it strictly increasing. For each k, you can make $\displaystyle x_{k^2+1}$ very slightly smaller, without disturbing the property that $\displaystyle y_{k^2+1} > y_{k^2}$. Then you can interpolate the values of $\displaystyle x_n$ linearly for $\displaystyle k^2+1\leqslant n\leqslant (k+1)^2$ so as to ensure that the sequence increases strictly.

that was brilliant!! thanks. it solved my question.
• Dec 29th 2011, 12:01 AM
abhishekkgp
Re: does the sequence necessarily become decreasing??
Quote:

Originally Posted by chisigma
If we define $\displaystyle \delta_{x}(n)= x_{n+1}-x_{n}$ then it must be...

a) $\displaystyle \delta_{x}(n)>0\ \forall n$

b) $\displaystyle \lim_{n \rightarrow \infty} x_{n}= x_{1} + \sum_{k=1}^{\infty} \delta_{x}(k)= x_{0}\le 1$ (1)

Now, if $\displaystyle y_{n}=\frac{x_{n}}{n}$ we define $\displaystyle \delta_{y}(n)= y_{n+1}-y_{n}$ and in few steps we find that...

$\displaystyle \delta_{y}(n)= \frac{\delta_{x}(n)-y_{n}}{n+1}$ (2)

Because the series in (1) converges, it must be necesserly for some $\displaystyle \varepsilon>0$ and $\displaystyle N$ $\displaystyle \delta_{x}(n)<\frac{\varepsilon}{n}\ \forall n>N$ and that means that $\displaystyle \delta_{x}(n)<y_{n}\ \forall n>N$ so that $\displaystyle \forall n>N$ is $\displaystyle \delta_{y}(n)<0$...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

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.
• Dec 29th 2011, 12:08 AM
Opalg
Re: does the sequence necessarily become decreasing??
Quote:

Originally Posted by chisigma
$\displaystyle x_{1} + \sum_{k=1}^{\infty} \delta_{x}(k)= x_{0}\le 1$ (1)

...

Because the series in (1) converges, it must be necessarly for some $\displaystyle \varepsilon>0$ and $\displaystyle N$ $\displaystyle \delta_{x}(n)<\frac{\varepsilon}{n}\ \forall n>N$

That is not true. Suppose for example that $\displaystyle a_n$ is always zero unless n is a perfect cube, with

$\displaystyle a_n = \begin{cases}1/\sqrt n = k^{-3/2} & \text{if }n=k^3, \\ 0 & \text{otherwise.} \end{cases}$

Then $\displaystyle \textstyle\sum a_n = \textstyle\sum k^{-3/2} < \infty$. But if $\displaystyle n=k^3$ then $\displaystyle na_n = k^{3/2} \to\infty$ as $\displaystyle k\to\infty.$ Therefore there do not exist $\displaystyle \varepsilon>0$ and $\displaystyle N$ such that $\displaystyle a_n<\frac{\varepsilon}{n}$ for all $\displaystyle n>N.$

The point there, as in the OP's problem, is that a series or sequence can have bad behaviour sporadically while still being well-behaved as a whole.
• Dec 29th 2011, 09:17 PM
chisigma
Re: does the sequence necessarily become decreasing??
Quote:

Originally Posted by chisigma
If we define $\displaystyle \delta_{x}(n)= x_{n+1}-x_{n}$ then it must be...

a) $\displaystyle \delta_{x}(n)>0\ \forall n$

b) $\displaystyle \lim_{n \rightarrow \infty} x_{n}= x_{1} + \sum_{k=1}^{\infty} \delta_{x}(k)= x_{0}\le 1$ (1)

Now, if $\displaystyle y_{n}=\frac{x_{n}}{n}$ we define $\displaystyle \delta_{y}(n)= y_{n+1}-y_{n}$ and in few steps we find that...

$\displaystyle \delta_{y}(n)= \frac{\delta_{x}(n)-y_{n}}{n+1}$ (2)

Because the series in (1) converges, it must be necesserly for some $\displaystyle \varepsilon>0$ and $\displaystyle N$ $\displaystyle \delta_{x}(n)<\frac{\varepsilon}{n}\ \forall n>N$ and that means that $\displaystyle \delta_{x}(n)<y_{n}\ \forall n>N$ so that $\displaystyle \forall n>N$ is $\displaystyle \delta_{y}(n)<0$...

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

a) $\displaystyle \delta_{x}(n)>0\ \forall n$

b) $\displaystyle \lim_{n \rightarrow \infty} x_{n}= x_{1} + \sum_{k=1}^{\infty} \delta_{x}(k)= x_{0}\le 1$ (1)

... and the a) means that 'punctured infinite sums' like $\displaystyle \sum_{n} \delta_{n}$ where...

$\displaystyle \delta_{n}=\begin{cases} \frac{1}{\sqrt{n}} &\text{if}\ n=k^{3}\\ 0 &\text{otherwise}\end{cases}$ (2)

... aren't adequate counterexamples...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
• Dec 30th 2011, 12:28 AM
Opalg
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) $\displaystyle \delta_{x}(n)>0\ \forall n$

b) $\displaystyle \lim_{n \rightarrow \infty} x_{n}= x_{1} + \sum_{k=1}^{\infty} \delta_{x}(k)= x_{0}\le 1$ (1)

... and the a) means that 'punctured infinite sums' like $\displaystyle \sum_{n} \delta_{n}$ where...

$\displaystyle \delta_{n}=\begin{cases} \frac{1}{\sqrt{n}} &\text{if}\ n=k^{3}\\ 0 &\text{otherwise}\end{cases}$ (2)

... aren't adequate counterexamples...

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

$\displaystyle \delta_{n}=\begin{cases} \frac{1}{\sqrt{n}} &\text{if}\ n=k^{3}, \\ \frac1{n^2}&\text{otherwise}.\end{cases}$

The fact is that the convergence of the series $\displaystyle \textstyle\sum\delta_n$ does not imply that there exist $\displaystyle \varepsilon>0$ and $\displaystyle N$ such that $\displaystyle \delta_n<\frac{\varepsilon}{n}$ for all $\displaystyle n>N$.