How do I simplify this expression with a floor function?

[japplepie]

I would want to change this expression to another one that would have P easily accessible.

\frac{\left \lceil (n+1)^{P} \right \rceil}{\left \lceil n^{P} \right \rceil}

As an example of 'accessible'; given (n+1)^P / n^P, I can make P more accessible by using log(). Then it becomes P * (log(n+1)-log(n)), where P is just outside everything else and is easily accessible by just dividing by something.

 

[RUber]

I have seen two different approaches to this sort of problem.
1) Make a conversion inside the ceiling function, such as $e^{p \log (n+1) }$
2) Add in a dummy term 0<w<1 such that the ceiling (n+1)^p = (n+1)^p +w.
The second option often let's you get sufficiently close to the answer you need.

 

[japplepie]

I have seen two different approaches to this sort of problem.
1) Make a conversion inside the ceiling function, such as $e^{p \log (n+1) }$
2) Add in a dummy term 0<w<1 such that the ceiling (n+1)^p = (n+1)^p +w.
The second option often let's you get sufficiently close to the answer you need.

Regarding the 2nd option,

I don't understand how to bring out P in the expression, ((n+1)^P + w1) / (n^P + w0)

 

[RUber]

Do you have any information about P or n, $n \in \mathbb{N}?, P>0 \in \mathbb{R}?$
What is the outcome you are looking for? Is it a bound of some sort, or a relation?
You could say that ceiling(n+1)^p = (n+1+w)^p. w would still be bounded in [0,1), and would tend toward zero much more reliably than having it outside the exponent.

 

[japplepie]

Do you have any information about P or n, $n \in \mathbb{N}?, P>0 \in \mathbb{R}?$
What is the outcome you are looking for? Is it a bound of some sort, or a relation?
You could say that ceiling(n+1)^p = (n+1+w)^p. w would still be bounded in [0,1), and would tend toward zero much more reliably than having it outside the exponent.

P is a real number greater than 0
n is a natural number (including 0)

I'm trying to make a convergence acceleration formula for a (rounded up) sequence of numbers, then it eventually leads to this problem:

seq(n) = ceil ((n+1)^P) / ceil (n^P) , for any real P>0

which requires me to completely isolate P on its own to one side of an equation

Note:
seq(n) isn't the sequence that I'm trying to accelerate the convergence on, it's just derived from something else.

 

[RUber]

If you use $R = \frac{ (n+1+w )^p }{(n+v)^p} = \left( \frac{n+1+w }{n+v}\right)^p$
then, $\ln R = p \ln \left( \frac{n+1+w }{n+v}\right).$
So, just in using the range [0,1) for w,v you can bound it by:
$\ln \left(\frac{ n+1 }{n+1}\right) < \frac{\ln R}{p}< \ln \left(\frac{n+2 }{n}\right)$

Is that a set of bounds you can work with? Or do you need finer resolution on how quickly w and v go to zero so you can estimate how close to $p \ln \frac{n+1}{n}$ the rate really is?

 

[japplepie]

If you use $R = \frac{ (n+1+w )^p }{(n+v)^p} = \left( \frac{n+1+w }{n+v}\right)^p$
then, $\ln R = p \ln \left( \frac{n+1+w }{n+v}\right).$
So, just in using the range [0,1) for w,v you can bound it by:
$\ln \left(\frac{ n+1 }{n+1}\right) < \frac{\ln R}{p}< \ln \left(\frac{n+2 }{n}\right)$

Is that a set of bounds you can work with? Or do you need finer resolution on how quickly w and v go to zero so you can estimate how close to $p \ln \frac{n+1}{n}$ the rate really is?

I think the first step would actually be:

R= ((n+1)^P + w) / (n^P + v)

If it is, I have no idea how to go on from there..

 

[RUber]

Good point. Upon further investigation, the assumption that you could bring a small number inside the exponential fell apart pretty quickly.

However, staying with the bounding idea,
$\frac{ (n+1)^p }{n^p +1 } <\frac{ \lceil (n+1)^p \rceil }{ \lceil (n)^p \rceil } < \frac{ (n+1)^p +1}{n^p }$
Which letting $Q =\frac{ (n+1)^p }{ (n)^p }$ can be simplified to:
$\frac{Q}{1+ \frac{1}{n^p} } < R < Q + \frac{1}{n^p}$
The upper bound is pretty straightforward, the lower bound can be expressed as:
$\frac{Q}{1+ \frac{1}{n^p} } = Q - \frac{Q}{n^p} + \frac{Q}{n^{2p}}- \frac{Q}{n^{3p}} ...$

Is this perhaps a form you could use to estimate your error bounds? Everything is expressed as a power of p, but not in such a form that you can just take the log of it straight away.

 

[japplepie]

Good point. Upon further investigation, the assumption that you could bring a small number inside the exponential fell apart pretty quickly.

However, staying with the bounding idea,
$\frac{ (n+1)^p }{n^p +1 } <\frac{ \lceil (n+1)^p \rceil }{ \lceil (n)^p \rceil } < \frac{ (n+1)^p +1}{n^p }$
Which letting $Q =\frac{ (n+1)^p }{ (n)^p }$ can be simplified to:
$\frac{Q}{1+ \frac{1}{n^p} } < R < Q + \frac{1}{n^p}$
The upper bound is pretty straightforward, the lower bound can be expressed as:
$\frac{Q}{1+ \frac{1}{n^p} } = Q - \frac{Q}{n^p} + \frac{Q}{n^{2p}}- \frac{Q}{n^{3p}} ...$

Is this perhaps a form you could use to estimate your error bounds? Everything is expressed as a power of p, but not in such a form that you can just take the log of it straight away.

That's very clever; I think I would be able to use it.

Thanks for the help!

------------
Edit:
What I decided to do next is to solve for the minimum and maximum values of P using the 2 bounds.

I'm having trouble with solving for P in the expression:

\frac{(n+1)^{P}}{n^{P}+1}=X

Is this even solvable?

 

[RUber]

I can't seem to get right to a solution.

If p were an integer, there are ways to simplify it with polynomial methods.
Otherwise, you can discuss the limits as n or p are large or small.
For p close to zero and n > 0, this will tend toward 1/2.
For p>1, as n grows this will tend toward 1.
Note that for an integer p,
$(n+1)^p = \sum_{i=0}^p \begin{pmatrix} n\\i\end{pmatrix}n^i=n^p + 1 + \sum_{i=1}^{p-1} \begin{pmatrix} n\\i\end{pmatrix}n^i.$
So, you could simplify
$\frac{(n+1)^p}{n^p + 1} = 1 + \frac{\sum_{i=1}^{p-1} \begin{pmatrix} n\\i\end{pmatrix}n^i}{n^p + 1}.$

Generally, I would plot the log of convergence factor vs. n. The slope of this plot should be an approximation for your exponent for the convergence rate $e^x$, with x being a negative value.

 

[japplepie]

I can't seem to get right to a solution.

If p were an integer, there are ways to simplify it with polynomial methods.
Otherwise, you can discuss the limits as n or p are large or small.
For p close to zero and n > 0, this will tend toward 1/2.
For p>1, as n grows this will tend toward 1.
Note that for an integer p,
$(n+1)^p = \sum_{i=0}^p \begin{pmatrix} n\\i\end{pmatrix}n^i=n^p + 1 + \sum_{i=1}^{p-1} \begin{pmatrix} n\\i\end{pmatrix}n^i.$
So, you could simplify
$\frac{(n+1)^p}{n^p + 1} = 1 + \frac{\sum_{i=1}^{p-1} \begin{pmatrix} n\\i\end{pmatrix}n^i}{n^p + 1}.$

Generally, I would plot the log of convergence factor vs. n. The slope of this plot should be an approximation for your exponent for the convergence rate $e^x$, with x being a negative value.

Unfortunately, I have no idea about what you said (I don't have that much of a background in maths, sorry)

But I have been searching around to look for ways to solve this and I found that perturbation theory could solve algebraic problems that are not exactly solvable; would that work?

I would try it out myself but I don't know how to, all the examples I could find and (kinda) understand aren't applicable in my case.

 

[RUber]

The $\begin{pmatrix} n\\i\end{pmatrix}$ notation means the combination or "n choose i" operation which simplifies to:
$\begin{pmatrix} n\\i\end{pmatrix}=\frac{n!}{(n-i)!i!}$
I made a mistake in the original formula:

for integer p,
$(n+1)^p = \sum_{i=0}^p \begin{pmatrix} n\\i\end{pmatrix}n^i=n^p + 1 + \sum_{i=1}^{p-1} \begin{pmatrix} n\\i\end{pmatrix}n^i.$
So, you could simplify
$\frac{(n+1)^p}{n^p + 1} = 1 + \frac{\sum_{i=1}^{p-1} \begin{pmatrix} n\\i\end{pmatrix}n^i}{n^p + 1}.$

Should be
for integer p,
$(n+1)^p = \sum_{i=0}^p \begin{pmatrix} p\\i\end{pmatrix}n^i=n^p + 1 + \sum_{i=1}^{p-1} \begin{pmatrix} p\\i\end{pmatrix}n^i.$

If you are just looking to estimate, you could only keep the highest order term in the expansion:
$\frac{(n+1)^p}{n^p + 1} = 1 + \frac{\sum_{i=1}^{p-1} \begin{pmatrix} p\\i\end{pmatrix}n^i}{n^p + 1}\\ =1 +\frac{pn^{p-1}}{n^p}+\mathcal{O}(n^{-2}) \\ \approx 1+ \frac pn.$