Analysis Ratio Test

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Give an example of a sequence $(a_n)_n$ such that $|a_n+1/a_n|<1$ for all $ninmathbbN$, but where $(a_n)_n$ does not converge to $0$.



This question came up in a text book and I've no idea what the answer it, any help, sorry for not knowing how to type math equations!







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  • A link to a MathJax Guide for the future: math.meta.stackexchange.com/questions/5020/…
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    Aug 11 at 1:11














up vote
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down vote

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Give an example of a sequence $(a_n)_n$ such that $|a_n+1/a_n|<1$ for all $ninmathbbN$, but where $(a_n)_n$ does not converge to $0$.



This question came up in a text book and I've no idea what the answer it, any help, sorry for not knowing how to type math equations!







share|cite|improve this question






















  • A link to a MathJax Guide for the future: math.meta.stackexchange.com/questions/5020/…
    – zzuussee
    Aug 11 at 1:11












up vote
2
down vote

favorite









up vote
2
down vote

favorite











Give an example of a sequence $(a_n)_n$ such that $|a_n+1/a_n|<1$ for all $ninmathbbN$, but where $(a_n)_n$ does not converge to $0$.



This question came up in a text book and I've no idea what the answer it, any help, sorry for not knowing how to type math equations!







share|cite|improve this question














Give an example of a sequence $(a_n)_n$ such that $|a_n+1/a_n|<1$ for all $ninmathbbN$, but where $(a_n)_n$ does not converge to $0$.



This question came up in a text book and I've no idea what the answer it, any help, sorry for not knowing how to type math equations!









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edited Aug 11 at 1:17









zzuussee

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1,701420










asked Aug 11 at 0:55









Brad Scott

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  • A link to a MathJax Guide for the future: math.meta.stackexchange.com/questions/5020/…
    – zzuussee
    Aug 11 at 1:11
















  • A link to a MathJax Guide for the future: math.meta.stackexchange.com/questions/5020/…
    – zzuussee
    Aug 11 at 1:11















A link to a MathJax Guide for the future: math.meta.stackexchange.com/questions/5020/…
– zzuussee
Aug 11 at 1:11




A link to a MathJax Guide for the future: math.meta.stackexchange.com/questions/5020/…
– zzuussee
Aug 11 at 1:11










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Take $(a_n)_ninmathbbN$ s.t. $a_n=frac1n+frac12$ f.a. $ninmathbbN$. You have that $a_n+1<a_n$ f.a. $ninmathbbN$, i.e. that $(a_n)_ninmathbbN$ is strictly monotone decreasing(check this). Thus



$$fraca_n+1a_n<1$$



as all $a_n$ are positive but clearly $lim_ntoinftya_n=1/2$.



EDIT: To give some suggestion of how to come up with such examples: analyze you conditions. Note that in your case, if $a_n>0$ f.a. $ninmathbbN$, then $|fraca_n+1a_n|=fraca_n+1a_n$ and thus



$$|fraca_n+1a_n|<1Leftrightarrowfraca_n+1a_n<1Leftrightarrow a_n+1<a_n$$



Thus, under some assumptions, your condition is the same as requiring that the sequence decreases strictly monotone. From, this on, it might be simpler to come up with examples.






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    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    2
    down vote



    accepted










    Take $(a_n)_ninmathbbN$ s.t. $a_n=frac1n+frac12$ f.a. $ninmathbbN$. You have that $a_n+1<a_n$ f.a. $ninmathbbN$, i.e. that $(a_n)_ninmathbbN$ is strictly monotone decreasing(check this). Thus



    $$fraca_n+1a_n<1$$



    as all $a_n$ are positive but clearly $lim_ntoinftya_n=1/2$.



    EDIT: To give some suggestion of how to come up with such examples: analyze you conditions. Note that in your case, if $a_n>0$ f.a. $ninmathbbN$, then $|fraca_n+1a_n|=fraca_n+1a_n$ and thus



    $$|fraca_n+1a_n|<1Leftrightarrowfraca_n+1a_n<1Leftrightarrow a_n+1<a_n$$



    Thus, under some assumptions, your condition is the same as requiring that the sequence decreases strictly monotone. From, this on, it might be simpler to come up with examples.






    share|cite|improve this answer


























      up vote
      2
      down vote



      accepted










      Take $(a_n)_ninmathbbN$ s.t. $a_n=frac1n+frac12$ f.a. $ninmathbbN$. You have that $a_n+1<a_n$ f.a. $ninmathbbN$, i.e. that $(a_n)_ninmathbbN$ is strictly monotone decreasing(check this). Thus



      $$fraca_n+1a_n<1$$



      as all $a_n$ are positive but clearly $lim_ntoinftya_n=1/2$.



      EDIT: To give some suggestion of how to come up with such examples: analyze you conditions. Note that in your case, if $a_n>0$ f.a. $ninmathbbN$, then $|fraca_n+1a_n|=fraca_n+1a_n$ and thus



      $$|fraca_n+1a_n|<1Leftrightarrowfraca_n+1a_n<1Leftrightarrow a_n+1<a_n$$



      Thus, under some assumptions, your condition is the same as requiring that the sequence decreases strictly monotone. From, this on, it might be simpler to come up with examples.






      share|cite|improve this answer
























        up vote
        2
        down vote



        accepted







        up vote
        2
        down vote



        accepted






        Take $(a_n)_ninmathbbN$ s.t. $a_n=frac1n+frac12$ f.a. $ninmathbbN$. You have that $a_n+1<a_n$ f.a. $ninmathbbN$, i.e. that $(a_n)_ninmathbbN$ is strictly monotone decreasing(check this). Thus



        $$fraca_n+1a_n<1$$



        as all $a_n$ are positive but clearly $lim_ntoinftya_n=1/2$.



        EDIT: To give some suggestion of how to come up with such examples: analyze you conditions. Note that in your case, if $a_n>0$ f.a. $ninmathbbN$, then $|fraca_n+1a_n|=fraca_n+1a_n$ and thus



        $$|fraca_n+1a_n|<1Leftrightarrowfraca_n+1a_n<1Leftrightarrow a_n+1<a_n$$



        Thus, under some assumptions, your condition is the same as requiring that the sequence decreases strictly monotone. From, this on, it might be simpler to come up with examples.






        share|cite|improve this answer














        Take $(a_n)_ninmathbbN$ s.t. $a_n=frac1n+frac12$ f.a. $ninmathbbN$. You have that $a_n+1<a_n$ f.a. $ninmathbbN$, i.e. that $(a_n)_ninmathbbN$ is strictly monotone decreasing(check this). Thus



        $$fraca_n+1a_n<1$$



        as all $a_n$ are positive but clearly $lim_ntoinftya_n=1/2$.



        EDIT: To give some suggestion of how to come up with such examples: analyze you conditions. Note that in your case, if $a_n>0$ f.a. $ninmathbbN$, then $|fraca_n+1a_n|=fraca_n+1a_n$ and thus



        $$|fraca_n+1a_n|<1Leftrightarrowfraca_n+1a_n<1Leftrightarrow a_n+1<a_n$$



        Thus, under some assumptions, your condition is the same as requiring that the sequence decreases strictly monotone. From, this on, it might be simpler to come up with examples.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Aug 11 at 1:10

























        answered Aug 11 at 1:05









        zzuussee

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