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Calculating limit of second-order recursive sequence


(limit of) a linear second order recurrence relation with variable coefficientsProving the limit of a recursive sequenceRecursive equation with limitRecursive sequence with square rootCalculating limit of sequence by Euler $e$Finding the limits to sequences given by recursive transformation of a vector: $a_n = a_n-1M$How can I find an explicit expression for this recursively defined sequence?Calculate a limit of recursive sequenceLimit of a sequence with real and natural number variablesFinding the limit of a recursive complex sequence













2












$begingroup$


Can someone give me a hint on how to calculate the limit for the following second-order recursive sequence:



beginalign
a_0 &= c_0, \
a_1 &= c_1, \
a_n &=fraca_n-1+a_n-22-frac(a_n-1)^2+(a_n-2)^22+(fraca_n-1+a_n-22)^2
endalign



I simulated it in Excel and I know that it converges to a constant or goes to negative infinity depending on $c_0,c_1$.



Can I separate the limit into 3 parts and calculate them one by one? Is it correct that I can not, because not all limits for each part exist?



I am mostly interested in the formula for the limit dependent on $c_0, c_1$ for an unrelated theoretical argument.










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$endgroup$
















    2












    $begingroup$


    Can someone give me a hint on how to calculate the limit for the following second-order recursive sequence:



    beginalign
    a_0 &= c_0, \
    a_1 &= c_1, \
    a_n &=fraca_n-1+a_n-22-frac(a_n-1)^2+(a_n-2)^22+(fraca_n-1+a_n-22)^2
    endalign



    I simulated it in Excel and I know that it converges to a constant or goes to negative infinity depending on $c_0,c_1$.



    Can I separate the limit into 3 parts and calculate them one by one? Is it correct that I can not, because not all limits for each part exist?



    I am mostly interested in the formula for the limit dependent on $c_0, c_1$ for an unrelated theoretical argument.










    share|cite|improve this question









    New contributor




    F.L is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.







    $endgroup$














      2












      2








      2


      1



      $begingroup$


      Can someone give me a hint on how to calculate the limit for the following second-order recursive sequence:



      beginalign
      a_0 &= c_0, \
      a_1 &= c_1, \
      a_n &=fraca_n-1+a_n-22-frac(a_n-1)^2+(a_n-2)^22+(fraca_n-1+a_n-22)^2
      endalign



      I simulated it in Excel and I know that it converges to a constant or goes to negative infinity depending on $c_0,c_1$.



      Can I separate the limit into 3 parts and calculate them one by one? Is it correct that I can not, because not all limits for each part exist?



      I am mostly interested in the formula for the limit dependent on $c_0, c_1$ for an unrelated theoretical argument.










      share|cite|improve this question









      New contributor




      F.L is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.







      $endgroup$




      Can someone give me a hint on how to calculate the limit for the following second-order recursive sequence:



      beginalign
      a_0 &= c_0, \
      a_1 &= c_1, \
      a_n &=fraca_n-1+a_n-22-frac(a_n-1)^2+(a_n-2)^22+(fraca_n-1+a_n-22)^2
      endalign



      I simulated it in Excel and I know that it converges to a constant or goes to negative infinity depending on $c_0,c_1$.



      Can I separate the limit into 3 parts and calculate them one by one? Is it correct that I can not, because not all limits for each part exist?



      I am mostly interested in the formula for the limit dependent on $c_0, c_1$ for an unrelated theoretical argument.







      sequences-and-series limits recurrence-relations sequent-calculus






      share|cite|improve this question









      New contributor




      F.L is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      share|cite|improve this question









      New contributor




      F.L is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.









      share|cite|improve this question




      share|cite|improve this question








      edited Mar 10 at 20:59







      F.L













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      asked Mar 10 at 19:35









      F.LF.L

      112




      112




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      New contributor





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          1 Answer
          1






          active

          oldest

          votes


















          0












          $begingroup$

          When $a_1-a_2geq 6$ or $ a_2-a_1geq 4 $, it diverges. Other cases
          are convergent.



          Proof : i) $$a_n
          = fraca_n-1+a_n-22 - (fraca_n-1-a_n-2 2)^2 $$



          When $|a_1-a_2|geq 6$, then $|a_n-a_n-1|geq 6$. It does not converge.



          ii) $a_2<a_1$ : If $a_1-a_2<6$ is close to $6$, then
          $$ a_2-a_3 < a_1-a_2 $$



          Hence there is large $N$ s.t. $a_i>
          a_i+1$
          for $ileq N-2$ and $a_N-1<a_N$.



          iii) $a_1<a_2$ : If $a_2-a_1geq 4
          $
          , then $a_2-a_3geq 6$.



          If $a_2-a_1<4$, then $a_2-a_3<6$.






          share|cite|improve this answer









          $endgroup$












            Your Answer





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            1 Answer
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            1 Answer
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            0












            $begingroup$

            When $a_1-a_2geq 6$ or $ a_2-a_1geq 4 $, it diverges. Other cases
            are convergent.



            Proof : i) $$a_n
            = fraca_n-1+a_n-22 - (fraca_n-1-a_n-2 2)^2 $$



            When $|a_1-a_2|geq 6$, then $|a_n-a_n-1|geq 6$. It does not converge.



            ii) $a_2<a_1$ : If $a_1-a_2<6$ is close to $6$, then
            $$ a_2-a_3 < a_1-a_2 $$



            Hence there is large $N$ s.t. $a_i>
            a_i+1$
            for $ileq N-2$ and $a_N-1<a_N$.



            iii) $a_1<a_2$ : If $a_2-a_1geq 4
            $
            , then $a_2-a_3geq 6$.



            If $a_2-a_1<4$, then $a_2-a_3<6$.






            share|cite|improve this answer









            $endgroup$

















              0












              $begingroup$

              When $a_1-a_2geq 6$ or $ a_2-a_1geq 4 $, it diverges. Other cases
              are convergent.



              Proof : i) $$a_n
              = fraca_n-1+a_n-22 - (fraca_n-1-a_n-2 2)^2 $$



              When $|a_1-a_2|geq 6$, then $|a_n-a_n-1|geq 6$. It does not converge.



              ii) $a_2<a_1$ : If $a_1-a_2<6$ is close to $6$, then
              $$ a_2-a_3 < a_1-a_2 $$



              Hence there is large $N$ s.t. $a_i>
              a_i+1$
              for $ileq N-2$ and $a_N-1<a_N$.



              iii) $a_1<a_2$ : If $a_2-a_1geq 4
              $
              , then $a_2-a_3geq 6$.



              If $a_2-a_1<4$, then $a_2-a_3<6$.






              share|cite|improve this answer









              $endgroup$















                0












                0








                0





                $begingroup$

                When $a_1-a_2geq 6$ or $ a_2-a_1geq 4 $, it diverges. Other cases
                are convergent.



                Proof : i) $$a_n
                = fraca_n-1+a_n-22 - (fraca_n-1-a_n-2 2)^2 $$



                When $|a_1-a_2|geq 6$, then $|a_n-a_n-1|geq 6$. It does not converge.



                ii) $a_2<a_1$ : If $a_1-a_2<6$ is close to $6$, then
                $$ a_2-a_3 < a_1-a_2 $$



                Hence there is large $N$ s.t. $a_i>
                a_i+1$
                for $ileq N-2$ and $a_N-1<a_N$.



                iii) $a_1<a_2$ : If $a_2-a_1geq 4
                $
                , then $a_2-a_3geq 6$.



                If $a_2-a_1<4$, then $a_2-a_3<6$.






                share|cite|improve this answer









                $endgroup$



                When $a_1-a_2geq 6$ or $ a_2-a_1geq 4 $, it diverges. Other cases
                are convergent.



                Proof : i) $$a_n
                = fraca_n-1+a_n-22 - (fraca_n-1-a_n-2 2)^2 $$



                When $|a_1-a_2|geq 6$, then $|a_n-a_n-1|geq 6$. It does not converge.



                ii) $a_2<a_1$ : If $a_1-a_2<6$ is close to $6$, then
                $$ a_2-a_3 < a_1-a_2 $$



                Hence there is large $N$ s.t. $a_i>
                a_i+1$
                for $ileq N-2$ and $a_N-1<a_N$.



                iii) $a_1<a_2$ : If $a_2-a_1geq 4
                $
                , then $a_2-a_3geq 6$.



                If $a_2-a_1<4$, then $a_2-a_3<6$.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 2 days ago









                HK LeeHK Lee

                14.1k52360




                14.1k52360




















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