If $lim_{ntoinfty}a_n=0$, then $lim_{ntoinfty}a_n^n=0$ [closed]

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Let ${a_n}$ be a sequence such that $lim_{ntoinfty} a_n = 0$. Show that $lim_{ntoinfty} a_n^n = 0$.




I haven't been able to find anything about this question and would greatly appreciate if anyone knows how to do it.






sequences-and-series limits convergence infinity







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closed as off-topic by Nosrati, RRL, amWhy, Michael Hoppe, Cesareo Dec 5 '18 at 19:33


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Nosrati, RRL, amWhy, Michael Hoppe, Cesareo

If this question can be reworded to fit the rules in the help center, please edit the question.









  • 1




    What can you say about the result when a (real) number close to zero is raised to a "large" positive integer power?
    – hardmath
    Dec 5 '18 at 17:02
















-2















Let ${a_n}$ be a sequence such that $lim_{ntoinfty} a_n = 0$. Show that $lim_{ntoinfty} a_n^n = 0$.




I haven't been able to find anything about this question and would greatly appreciate if anyone knows how to do it.






sequences-and-series limits convergence infinity







share|cite|improve this question















closed as off-topic by Nosrati, RRL, amWhy, Michael Hoppe, Cesareo Dec 5 '18 at 19:33


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Nosrati, RRL, amWhy, Michael Hoppe, Cesareo

If this question can be reworded to fit the rules in the help center, please edit the question.









  • 1




    What can you say about the result when a (real) number close to zero is raised to a "large" positive integer power?
    – hardmath
    Dec 5 '18 at 17:02














-2












-2








-2








Let ${a_n}$ be a sequence such that $lim_{ntoinfty} a_n = 0$. Show that $lim_{ntoinfty} a_n^n = 0$.




I haven't been able to find anything about this question and would greatly appreciate if anyone knows how to do it.






sequences-and-series limits convergence infinity







share|cite|improve this question
















Let ${a_n}$ be a sequence such that $lim_{ntoinfty} a_n = 0$. Show that $lim_{ntoinfty} a_n^n = 0$.




I haven't been able to find anything about this question and would greatly appreciate if anyone knows how to do it.






sequences-and-series limits convergence infinity




sequences-and-series limits convergence infinity






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 5 '18 at 17:03









gt6989b

33.3k22452




33.3k22452










asked Dec 5 '18 at 16:58









Mich2908Mich2908

24




24




closed as off-topic by Nosrati, RRL, amWhy, Michael Hoppe, Cesareo Dec 5 '18 at 19:33


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Nosrati, RRL, amWhy, Michael Hoppe, Cesareo

If this question can be reworded to fit the rules in the help center, please edit the question.




closed as off-topic by Nosrati, RRL, amWhy, Michael Hoppe, Cesareo Dec 5 '18 at 19:33


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Nosrati, RRL, amWhy, Michael Hoppe, Cesareo

If this question can be reworded to fit the rules in the help center, please edit the question.








  • 1




    What can you say about the result when a (real) number close to zero is raised to a "large" positive integer power?
    – hardmath
    Dec 5 '18 at 17:02














  • 1




    What can you say about the result when a (real) number close to zero is raised to a "large" positive integer power?
    – hardmath
    Dec 5 '18 at 17:02








1




1




What can you say about the result when a (real) number close to zero is raised to a "large" positive integer power?
– hardmath
Dec 5 '18 at 17:02




What can you say about the result when a (real) number close to zero is raised to a "large" positive integer power?
– hardmath
Dec 5 '18 at 17:02










2 Answers
2






active

oldest

votes


















1














$a_nto 0Rightarrow forall epsilon>0,exists n_0inmathbb{N}: |a_n|<epsilon forall n>n_0$



For $epsilon=1,exists n_oinmathbb{N}: |a_n|<1 forall n>n_0 Rightarrow |a_n^n|to 0
Rightarrow a_n^nto 0$






share|cite|improve this answer





























    0














    HINT



    Use the Root test on the sequence $b_n = a_n^n$.






    share|cite|improve this answer




























      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      1














      $a_nto 0Rightarrow forall epsilon>0,exists n_0inmathbb{N}: |a_n|<epsilon forall n>n_0$



      For $epsilon=1,exists n_oinmathbb{N}: |a_n|<1 forall n>n_0 Rightarrow |a_n^n|to 0
      Rightarrow a_n^nto 0$






      share|cite|improve this answer


























        1














        $a_nto 0Rightarrow forall epsilon>0,exists n_0inmathbb{N}: |a_n|<epsilon forall n>n_0$



        For $epsilon=1,exists n_oinmathbb{N}: |a_n|<1 forall n>n_0 Rightarrow |a_n^n|to 0
        Rightarrow a_n^nto 0$






        share|cite|improve this answer
























          1












          1








          1






          $a_nto 0Rightarrow forall epsilon>0,exists n_0inmathbb{N}: |a_n|<epsilon forall n>n_0$



          For $epsilon=1,exists n_oinmathbb{N}: |a_n|<1 forall n>n_0 Rightarrow |a_n^n|to 0
          Rightarrow a_n^nto 0$






          share|cite|improve this answer












          $a_nto 0Rightarrow forall epsilon>0,exists n_0inmathbb{N}: |a_n|<epsilon forall n>n_0$



          For $epsilon=1,exists n_oinmathbb{N}: |a_n|<1 forall n>n_0 Rightarrow |a_n^n|to 0
          Rightarrow a_n^nto 0$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 5 '18 at 17:03









          giannispapavgiannispapav

          1,534324




          1,534324























              0














              HINT



              Use the Root test on the sequence $b_n = a_n^n$.






              share|cite|improve this answer


























                0














                HINT



                Use the Root test on the sequence $b_n = a_n^n$.






                share|cite|improve this answer
























                  0












                  0








                  0






                  HINT



                  Use the Root test on the sequence $b_n = a_n^n$.






                  share|cite|improve this answer












                  HINT



                  Use the Root test on the sequence $b_n = a_n^n$.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Dec 5 '18 at 17:02









                  gt6989bgt6989b

                  33.3k22452




                  33.3k22452















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