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Math Help - Solving equation with parts of a sum and a sum

  1. #1
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    Question Solving equation with parts of a sum and a sum

    Is it possible to solve any of these equations and thus find \Delta x_{n} or \sum_{0}^{n}\Delta x_{n}?

    Define

    \Delta x_{0}=V\frac{l}{v_{0}}

    and

    \Delta x_{1}=V\frac{l-\Delta x_{0}}{v_{0}+V}

    and

    \Delta x_{n+1}=V\frac{l-\sum_{0}^{n}\Delta x_{n}}{v_{0}+nV}.

    Since V\ll v_{0} this can be simplified(?) to

    \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.
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  2. #2
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    Re: Solving equation with parts of a sum and a sum

    Recurrence relations can sometimes be solved using a Z transform.
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  3. #3
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    Re: Solving equation with parts of a sum and a sum

    Quote Originally Posted by fysikbengt View Post
    Is it possible to solve any of these equations and thus find \Delta x_{n} or \sum_{0}^{n}\Delta x_{n}?

    Define

    \Delta x_{0}=V\frac{l}{v_{0}}

    and

    \Delta x_{1}=V\frac{l-\Delta x_{0}}{v_{0}+V}

    and

    \Delta x_{n+1}=V\frac{l-\sum_{0}^{n}\Delta x_{n}}{v_{0}+nV}.

    Since V\ll v_{0} this can be simplified(?) to

    \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,

    \Delta x_{n+1}=V\left(\frac{l-\displaystyle\sum_{0}^{n}\Delta x_{n}}{v_{0}+nV}\right)

    Therefore, \Delta x_{n}=V\left(\frac{l-\displaystyle\sum_{0}^{n-1}\Delta x_{n}}{v_{0}+(n-1)V}\right)

    \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, V<<v_0~;~ v_0+nV\approx{v_0+(n-1)V}

    \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.
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  4. #4
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    Re: Solving equation with parts of a sum and a sum

    Quote Originally Posted by Sudharaka View Post
    Dear fysikbengt,

    \Delta x_{n+1}=V\left(\frac{l-\displaystyle\sum_{0}^{n}\Delta x_{n}}{v_{0}+nV}\right)

    Therefore, \Delta x_{n}=V\left(\frac{l-\displaystyle\sum_{0}^{n-1}\Delta x_{n}}{v_{0}+(n-1)V}\right)

    \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, V<<v_0~;~ v_0+nV\approx{v_0+(n-1)V}

    \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.
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  5. #5
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    Re: Solving equation with parts of a sum and a sum

    Quote Originally Posted by ojones View Post
    Recurrence relations can sometimes be solved using a Z transform.
    I tried to study 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.
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