Example of sequences ${x_n}$ and ${y_n}$ with matching ranges of values but different limits.












2















Find an example of two sequences ${x_n}$ and ${y_n}$ such that $R(x_n) = R(y_n)$ and:





  1. $x_n$ and $y_n$ are convergent, and $lim_{n to infty} x_n ne lim_{n to infty} y_n$;






  1. $x_n$ converges and $y_n$ diverges.


Here $R(x)$ denotes the set of all possible values of the sequence. Roughly speaking $y_n$ is a permutation of $x_n$ (at least as far as I understood it).



For the first case I was thinking of such a function which would have the same range. For example consider the following sequences:



$$
f(x) = arcsinleft(frac{x-2}{x-1}right) \
y(x) = arcsinleft(-frac{x-2}{x-1}right)
$$



Clearly both of them have the same range of values but their limits are different. The problem is that replacing $x$ with $n$ breaks continuity and we get this.



Second case is unclear to me.



Could we find some sequences satisfying $(1)$ and $(2)$?






calculus sequences-and-series limits examples-counterexamples







share|cite|improve this question
























  • A general strategy for looking for examples would be to hope that there is an example of the simplest possible type to narrow down the search space, and gradually ratcheting up the complexity as needed. Can you find an example where the ranges have size 1? If not, how about size 2? If not,...
    – Mark S.
    Nov 25 at 14:02










  • What about {1, -1, -1, -1, ...} and {-1, 1, 1, 1, ...} or is that not what you mean by R?
    – Paul
    Nov 25 at 14:03
















2















Find an example of two sequences ${x_n}$ and ${y_n}$ such that $R(x_n) = R(y_n)$ and:





  1. $x_n$ and $y_n$ are convergent, and $lim_{n to infty} x_n ne lim_{n to infty} y_n$;






  1. $x_n$ converges and $y_n$ diverges.


Here $R(x)$ denotes the set of all possible values of the sequence. Roughly speaking $y_n$ is a permutation of $x_n$ (at least as far as I understood it).



For the first case I was thinking of such a function which would have the same range. For example consider the following sequences:



$$
f(x) = arcsinleft(frac{x-2}{x-1}right) \
y(x) = arcsinleft(-frac{x-2}{x-1}right)
$$



Clearly both of them have the same range of values but their limits are different. The problem is that replacing $x$ with $n$ breaks continuity and we get this.



Second case is unclear to me.



Could we find some sequences satisfying $(1)$ and $(2)$?






calculus sequences-and-series limits examples-counterexamples







share|cite|improve this question
























  • A general strategy for looking for examples would be to hope that there is an example of the simplest possible type to narrow down the search space, and gradually ratcheting up the complexity as needed. Can you find an example where the ranges have size 1? If not, how about size 2? If not,...
    – Mark S.
    Nov 25 at 14:02










  • What about {1, -1, -1, -1, ...} and {-1, 1, 1, 1, ...} or is that not what you mean by R?
    – Paul
    Nov 25 at 14:03














2












2








2








Find an example of two sequences ${x_n}$ and ${y_n}$ such that $R(x_n) = R(y_n)$ and:





  1. $x_n$ and $y_n$ are convergent, and $lim_{n to infty} x_n ne lim_{n to infty} y_n$;






  1. $x_n$ converges and $y_n$ diverges.


Here $R(x)$ denotes the set of all possible values of the sequence. Roughly speaking $y_n$ is a permutation of $x_n$ (at least as far as I understood it).



For the first case I was thinking of such a function which would have the same range. For example consider the following sequences:



$$
f(x) = arcsinleft(frac{x-2}{x-1}right) \
y(x) = arcsinleft(-frac{x-2}{x-1}right)
$$



Clearly both of them have the same range of values but their limits are different. The problem is that replacing $x$ with $n$ breaks continuity and we get this.



Second case is unclear to me.



Could we find some sequences satisfying $(1)$ and $(2)$?






calculus sequences-and-series limits examples-counterexamples







share|cite|improve this question
















Find an example of two sequences ${x_n}$ and ${y_n}$ such that $R(x_n) = R(y_n)$ and:





  1. $x_n$ and $y_n$ are convergent, and $lim_{n to infty} x_n ne lim_{n to infty} y_n$;






  1. $x_n$ converges and $y_n$ diverges.


Here $R(x)$ denotes the set of all possible values of the sequence. Roughly speaking $y_n$ is a permutation of $x_n$ (at least as far as I understood it).



For the first case I was thinking of such a function which would have the same range. For example consider the following sequences:



$$
f(x) = arcsinleft(frac{x-2}{x-1}right) \
y(x) = arcsinleft(-frac{x-2}{x-1}right)
$$



Clearly both of them have the same range of values but their limits are different. The problem is that replacing $x$ with $n$ breaks continuity and we get this.



Second case is unclear to me.



Could we find some sequences satisfying $(1)$ and $(2)$?






calculus sequences-and-series limits examples-counterexamples




calculus sequences-and-series limits examples-counterexamples






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




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edited Nov 25 at 14:13









amWhy

191k28224439




191k28224439










asked Nov 25 at 13:51









roman

1,71021120




1,71021120












  • A general strategy for looking for examples would be to hope that there is an example of the simplest possible type to narrow down the search space, and gradually ratcheting up the complexity as needed. Can you find an example where the ranges have size 1? If not, how about size 2? If not,...
    – Mark S.
    Nov 25 at 14:02










  • What about {1, -1, -1, -1, ...} and {-1, 1, 1, 1, ...} or is that not what you mean by R?
    – Paul
    Nov 25 at 14:03


















  • A general strategy for looking for examples would be to hope that there is an example of the simplest possible type to narrow down the search space, and gradually ratcheting up the complexity as needed. Can you find an example where the ranges have size 1? If not, how about size 2? If not,...
    – Mark S.
    Nov 25 at 14:02










  • What about {1, -1, -1, -1, ...} and {-1, 1, 1, 1, ...} or is that not what you mean by R?
    – Paul
    Nov 25 at 14:03
















A general strategy for looking for examples would be to hope that there is an example of the simplest possible type to narrow down the search space, and gradually ratcheting up the complexity as needed. Can you find an example where the ranges have size 1? If not, how about size 2? If not,...
– Mark S.
Nov 25 at 14:02




A general strategy for looking for examples would be to hope that there is an example of the simplest possible type to narrow down the search space, and gradually ratcheting up the complexity as needed. Can you find an example where the ranges have size 1? If not, how about size 2? If not,...
– Mark S.
Nov 25 at 14:02












What about {1, -1, -1, -1, ...} and {-1, 1, 1, 1, ...} or is that not what you mean by R?
– Paul
Nov 25 at 14:03




What about {1, -1, -1, -1, ...} and {-1, 1, 1, 1, ...} or is that not what you mean by R?
– Paul
Nov 25 at 14:03










2 Answers
2






active

oldest

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5














No need to consider such complicated functions. For (1), just take
$$ (x_n)_{n=1}^infty = (1,0,0, 0,ldots), quad (y_n)_{n=1}^infty = (0,1,1,1,ldots), $$
and for (2), keep the same $(x_n)$ and take
$$ (y_n)_{n=1}^infty = (1,0,1,0,ldots). $$






share|cite|improve this answer





























    1














    Example for Case 1:

    Let $x_0 = -1, x_n=1$ for $n geq 1$

    Let $y_0 = 1, y_n=-1$ for $n geq 1$



    Example for Case 2:


    Let $x_0 = -1, x_n=1$ for $n geq 1$

    Let $y_n = (-1)^n$






    share|cite|improve this answer





















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      2 Answers
      2






      active

      oldest

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      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

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      active

      oldest

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      5














      No need to consider such complicated functions. For (1), just take
      $$ (x_n)_{n=1}^infty = (1,0,0, 0,ldots), quad (y_n)_{n=1}^infty = (0,1,1,1,ldots), $$
      and for (2), keep the same $(x_n)$ and take
      $$ (y_n)_{n=1}^infty = (1,0,1,0,ldots). $$






      share|cite|improve this answer


























        5














        No need to consider such complicated functions. For (1), just take
        $$ (x_n)_{n=1}^infty = (1,0,0, 0,ldots), quad (y_n)_{n=1}^infty = (0,1,1,1,ldots), $$
        and for (2), keep the same $(x_n)$ and take
        $$ (y_n)_{n=1}^infty = (1,0,1,0,ldots). $$






        share|cite|improve this answer
























          5












          5








          5






          No need to consider such complicated functions. For (1), just take
          $$ (x_n)_{n=1}^infty = (1,0,0, 0,ldots), quad (y_n)_{n=1}^infty = (0,1,1,1,ldots), $$
          and for (2), keep the same $(x_n)$ and take
          $$ (y_n)_{n=1}^infty = (1,0,1,0,ldots). $$






          share|cite|improve this answer












          No need to consider such complicated functions. For (1), just take
          $$ (x_n)_{n=1}^infty = (1,0,0, 0,ldots), quad (y_n)_{n=1}^infty = (0,1,1,1,ldots), $$
          and for (2), keep the same $(x_n)$ and take
          $$ (y_n)_{n=1}^infty = (1,0,1,0,ldots). $$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 25 at 14:02









          MisterRiemann

          5,7291624




          5,7291624























              1














              Example for Case 1:

              Let $x_0 = -1, x_n=1$ for $n geq 1$

              Let $y_0 = 1, y_n=-1$ for $n geq 1$



              Example for Case 2:


              Let $x_0 = -1, x_n=1$ for $n geq 1$

              Let $y_n = (-1)^n$






              share|cite|improve this answer


























                1














                Example for Case 1:

                Let $x_0 = -1, x_n=1$ for $n geq 1$

                Let $y_0 = 1, y_n=-1$ for $n geq 1$



                Example for Case 2:


                Let $x_0 = -1, x_n=1$ for $n geq 1$

                Let $y_n = (-1)^n$






                share|cite|improve this answer
























                  1












                  1








                  1






                  Example for Case 1:

                  Let $x_0 = -1, x_n=1$ for $n geq 1$

                  Let $y_0 = 1, y_n=-1$ for $n geq 1$



                  Example for Case 2:


                  Let $x_0 = -1, x_n=1$ for $n geq 1$

                  Let $y_n = (-1)^n$






                  share|cite|improve this answer












                  Example for Case 1:

                  Let $x_0 = -1, x_n=1$ for $n geq 1$

                  Let $y_0 = 1, y_n=-1$ for $n geq 1$



                  Example for Case 2:


                  Let $x_0 = -1, x_n=1$ for $n geq 1$

                  Let $y_n = (-1)^n$







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Nov 25 at 14:03









                  Naweed G. Seldon

                  1,272419




                  1,272419






























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