Solving equation with parts of a sum and a sum
Is it possible to solve any of these equations and thus find $\displaystyle \Delta x_{n}$ or $\displaystyle \sum_{0}^{n}\Delta x_{n}$?
Define
$\displaystyle \Delta x_{0}=V\frac{l}{v_{0}}$
and
$\displaystyle \Delta x_{1}=V\frac{l-\Delta x_{0}}{v_{0}+V}$
and
$\displaystyle \Delta x_{n+1}=V\frac{l-\sum_{0}^{n}\Delta x_{n}}{v_{0}+nV}$.
Since $\displaystyle V\ll v_{0}$ this can be simplified(?) to
$\displaystyle \Delta x_{n+1}-\Delta x_{n}=\frac{V \Delta x_{n}}{v_{0}+nV}$.
Maybe this euation is easier to solve. I cannot get any further.
Re: Solving equation with parts of a sum and a sum
Recurrence relations can sometimes be solved using a $\displaystyle Z$ transform.
Re: Solving equation with parts of a sum and a sum
Quote:
Originally Posted by
fysikbengt
Is it possible to solve any of these equations and thus find $\displaystyle \Delta x_{n}$ or $\displaystyle \sum_{0}^{n}\Delta x_{n}$?
Define
$\displaystyle \Delta x_{0}=V\frac{l}{v_{0}}$
and
$\displaystyle \Delta x_{1}=V\frac{l-\Delta x_{0}}{v_{0}+V}$
and
$\displaystyle \Delta x_{n+1}=V\frac{l-\sum_{0}^{n}\Delta x_{n}}{v_{0}+nV}$.
Since $\displaystyle V\ll v_{0}$ this can be simplified(?) to
$\displaystyle \Delta x_{n+1}-\Delta x_{n}=\frac{V \Delta x_{n}}{v_{0}+nV}$.
Maybe this euation is easier to solve. I cannot get any further.
Dear fysikbengt,
$\displaystyle \Delta x_{n+1}=V\left(\frac{l-\displaystyle\sum_{0}^{n}\Delta x_{n}}{v_{0}+nV}\right)$
Therefore, $\displaystyle \Delta x_{n}=V\left(\frac{l-\displaystyle\sum_{0}^{n-1}\Delta x_{n}}{v_{0}+(n-1)V}\right)$
$\displaystyle \Delta x_{n+1}-\Delta x_{n}=V\left(\frac{l-\displaystyle\sum_{0}^{n}\Delta x_{n}}{v_{0}+nV}\right)-V\left(\frac{l-\displaystyle\sum_{0}^{n-1}\Delta x_{n}}{v_{0}+(n-1)V}\right)$
Since, $\displaystyle V<<v_0~;~$$\displaystyle v_0+nV\approx{v_0+(n-1)V}$
$\displaystyle \Delta x_{n+1}-\Delta x_{n}=V\left(\frac{l-\displaystyle\sum_{0}^{n}\Delta x_{n}}{v_{0}+nV}\right)-V\left(\frac{l-\displaystyle\sum_{0}^{n-1}\Delta x_{n}}{v_{0}+nV}\right)=-\left(\frac{V \Delta x_{n}}{v_{0}+nV}\right)$
So I think you should have a the minus sign for your last expression.
Re: Solving equation with parts of a sum and a sum
Quote:
Originally Posted by
Sudharaka
Dear fysikbengt,
$\displaystyle \Delta x_{n+1}=V\left(\frac{l-\displaystyle\sum_{0}^{n}\Delta x_{n}}{v_{0}+nV}\right)$
Therefore, $\displaystyle \Delta x_{n}=V\left(\frac{l-\displaystyle\sum_{0}^{n-1}\Delta x_{n}}{v_{0}+(n-1)V}\right)$
$\displaystyle \Delta x_{n+1}-\Delta x_{n}=V\left(\frac{l-\displaystyle\sum_{0}^{n}\Delta x_{n}}{v_{0}+nV}\right)-V\left(\frac{l-\displaystyle\sum_{0}^{n-1}\Delta x_{n}}{v_{0}+(n-1)V}\right)$
Since, $\displaystyle V<<v_0~;~$$\displaystyle v_0+nV\approx{v_0+(n-1)V}$
$\displaystyle \Delta x_{n+1}-\Delta x_{n}=V\left(\frac{l-\displaystyle\sum_{0}^{n}\Delta x_{n}}{v_{0}+nV}\right)-V\left(\frac{l-\displaystyle\sum_{0}^{n-1}\Delta x_{n}}{v_{0}+nV}\right)=-\left(\frac{V \Delta x_{n}}{v_{0}+nV}\right)$
So I think you should have a the minus sign for your last expression.
Yes, of course the relation is convergent. The minus sign is there in my notes, it is a typo. Thanks for noticing.
Re: Solving equation with parts of a sum and a sum
Quote:
Originally Posted by
ojones
Recurrence relations can sometimes be solved using a $\displaystyle Z$ transform.
I tried to study $\displaystyle Z$ transforms and realised it is a full field not easily mastered. I have even forgot most of what I have read about Fourier transforms. So, I might just as well give up. This is not about laziness I hope.