A series that converges but the square of the terms of the series diverges

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I have to give an example of a convergent series $sum a_n$ for which $sum a_n^2 $ diverges.



I think that such a series cannot exist because if $sum a_n$ converges absolutely then $sum a_n^2 $ will always converge right?










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  • 1




    The implication you said after "because" is true but it's still possible that $sum_n a_n$ converges but not absolutely and $sum_n a_n^2$ does not converge (absolutely nor by any means).
    – Saucy O'Path
    26 mins ago











  • @max_zorn That's if $sum a_n$ converges absolutely.
    – Robert Israel
    22 mins ago










  • Yes, you are right Robert, sorry!
    – max_zorn
    18 mins ago














up vote
2
down vote

favorite












I have to give an example of a convergent series $sum a_n$ for which $sum a_n^2 $ diverges.



I think that such a series cannot exist because if $sum a_n$ converges absolutely then $sum a_n^2 $ will always converge right?










share|cite|improve this question







New contributor




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















  • 1




    The implication you said after "because" is true but it's still possible that $sum_n a_n$ converges but not absolutely and $sum_n a_n^2$ does not converge (absolutely nor by any means).
    – Saucy O'Path
    26 mins ago











  • @max_zorn That's if $sum a_n$ converges absolutely.
    – Robert Israel
    22 mins ago










  • Yes, you are right Robert, sorry!
    – max_zorn
    18 mins ago












up vote
2
down vote

favorite









up vote
2
down vote

favorite











I have to give an example of a convergent series $sum a_n$ for which $sum a_n^2 $ diverges.



I think that such a series cannot exist because if $sum a_n$ converges absolutely then $sum a_n^2 $ will always converge right?










share|cite|improve this question







New contributor




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











I have to give an example of a convergent series $sum a_n$ for which $sum a_n^2 $ diverges.



I think that such a series cannot exist because if $sum a_n$ converges absolutely then $sum a_n^2 $ will always converge right?







real-analysis sequences-and-series






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  • 1




    The implication you said after "because" is true but it's still possible that $sum_n a_n$ converges but not absolutely and $sum_n a_n^2$ does not converge (absolutely nor by any means).
    – Saucy O'Path
    26 mins ago











  • @max_zorn That's if $sum a_n$ converges absolutely.
    – Robert Israel
    22 mins ago










  • Yes, you are right Robert, sorry!
    – max_zorn
    18 mins ago












  • 1




    The implication you said after "because" is true but it's still possible that $sum_n a_n$ converges but not absolutely and $sum_n a_n^2$ does not converge (absolutely nor by any means).
    – Saucy O'Path
    26 mins ago











  • @max_zorn That's if $sum a_n$ converges absolutely.
    – Robert Israel
    22 mins ago










  • Yes, you are right Robert, sorry!
    – max_zorn
    18 mins ago







1




1




The implication you said after "because" is true but it's still possible that $sum_n a_n$ converges but not absolutely and $sum_n a_n^2$ does not converge (absolutely nor by any means).
– Saucy O'Path
26 mins ago





The implication you said after "because" is true but it's still possible that $sum_n a_n$ converges but not absolutely and $sum_n a_n^2$ does not converge (absolutely nor by any means).
– Saucy O'Path
26 mins ago













@max_zorn That's if $sum a_n$ converges absolutely.
– Robert Israel
22 mins ago




@max_zorn That's if $sum a_n$ converges absolutely.
– Robert Israel
22 mins ago












Yes, you are right Robert, sorry!
– max_zorn
18 mins ago




Yes, you are right Robert, sorry!
– max_zorn
18 mins ago










4 Answers
4






active

oldest

votes

















up vote
3
down vote













The alternating series test gives a wealth of examples. Take
$$
a_n=(-1)^n/sqrtn
$$

for example.






share|cite|improve this answer




















  • You beat me to it by $14$ seconds.
    – Robert Israel
    24 mins ago

















up vote
3
down vote













Try $$sum_n=1^infty frac(-1)^nsqrtn$$






share|cite|improve this answer



























    up vote
    0
    down vote













    The simplest example I have in mind is :
    $$sum_n=1^+inftyfrac(-1)^nsqrtn $$



    Beware that convergent (CV) and absolutely convergent (ACV) can be very different.
    Indeed, if $sum_n=1^+infty a_n $ is ACV then, $sum_n=1^+infty a_n^2$ is also ACV. To prove it, you can simply notice that, since $lim a_n=0$ then for $n$ big enough, $|a_n^2|<|a_n|$.



    You can also ask yourself a more general question :
    What are the function $f:mathbb R rightarrow mathbb R$ such that for all
    $sum a_n$ CV (resp. ACV), $sum f(a_n)$ is CV (resp. ACV).



    It is a (difficult) exercise to show that for $f:mathbb Rrightarrow mathbb R$,



    $$sum a_n ~CV Rightarrow sum f(a_n) ~CV quad textiff quad exists eta>0,exists lambdain mathbb R,forall xin ]-eta,eta[, quad f(x)=lambda x $$



    $$sum a_n ~CV Rightarrow sum f(a_n) ~ACV quad textiff quad exists eta>0,forall xin ]-eta,eta[, quad f(x)=0 quadquadquad~~$$



    $$sum a_n ~ACV Rightarrow sum f(a_n) ~ACV quad textiff quadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquad$$ $$quad f(0)=0 text and exists eta>0,exists M >0,forall xin ]-eta,eta[, |f(x)|leq M |x| $$



    $$sum a_n ACV Rightarrow sum f(a_n) ~CV quad textiff quad sum a_n ~ACV Rightarrow sum f(a_n) ~ACV~~~~~~~~~~~~~~ $$






    share|cite|improve this answer










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    • 1




      $sum_n a_n^2$ converges for this one.
      – Robert Israel
      23 mins ago










    • oups I answer too fast ^^
      – Can I play with Mathness
      20 mins ago










    • Can you correct it? Want to upvote the distinction between ACV and CV but can't bring myself to do it with the error
      – Oscar Lanzi
      11 mins ago










    • corrected, I also added a more general statement.
      – Can I play with Mathness
      10 mins ago

















    up vote
    0
    down vote













    More generally,
    if $a_n = dfrac(-1)^nn^1/(2m)$,
    then
    $sum_n=1^infty a_n
    $

    converges for integer $m ge 0$
    and
    $sum_n=1^infty a_n^2m
    $

    diverges.






    share|cite




















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      4 Answers
      4






      active

      oldest

      votes








      4 Answers
      4






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes








      up vote
      3
      down vote













      The alternating series test gives a wealth of examples. Take
      $$
      a_n=(-1)^n/sqrtn
      $$

      for example.






      share|cite|improve this answer




















      • You beat me to it by $14$ seconds.
        – Robert Israel
        24 mins ago














      up vote
      3
      down vote













      The alternating series test gives a wealth of examples. Take
      $$
      a_n=(-1)^n/sqrtn
      $$

      for example.






      share|cite|improve this answer




















      • You beat me to it by $14$ seconds.
        – Robert Israel
        24 mins ago












      up vote
      3
      down vote










      up vote
      3
      down vote









      The alternating series test gives a wealth of examples. Take
      $$
      a_n=(-1)^n/sqrtn
      $$

      for example.






      share|cite|improve this answer












      The alternating series test gives a wealth of examples. Take
      $$
      a_n=(-1)^n/sqrtn
      $$

      for example.







      share|cite|improve this answer












      share|cite|improve this answer



      share|cite|improve this answer










      answered 25 mins ago









      Foobaz John

      19.1k41247




      19.1k41247











      • You beat me to it by $14$ seconds.
        – Robert Israel
        24 mins ago
















      • You beat me to it by $14$ seconds.
        – Robert Israel
        24 mins ago















      You beat me to it by $14$ seconds.
      – Robert Israel
      24 mins ago




      You beat me to it by $14$ seconds.
      – Robert Israel
      24 mins ago










      up vote
      3
      down vote













      Try $$sum_n=1^infty frac(-1)^nsqrtn$$






      share|cite|improve this answer
























        up vote
        3
        down vote













        Try $$sum_n=1^infty frac(-1)^nsqrtn$$






        share|cite|improve this answer






















          up vote
          3
          down vote










          up vote
          3
          down vote









          Try $$sum_n=1^infty frac(-1)^nsqrtn$$






          share|cite|improve this answer












          Try $$sum_n=1^infty frac(-1)^nsqrtn$$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 25 mins ago









          Robert Israel

          312k23204450




          312k23204450




















              up vote
              0
              down vote













              The simplest example I have in mind is :
              $$sum_n=1^+inftyfrac(-1)^nsqrtn $$



              Beware that convergent (CV) and absolutely convergent (ACV) can be very different.
              Indeed, if $sum_n=1^+infty a_n $ is ACV then, $sum_n=1^+infty a_n^2$ is also ACV. To prove it, you can simply notice that, since $lim a_n=0$ then for $n$ big enough, $|a_n^2|<|a_n|$.



              You can also ask yourself a more general question :
              What are the function $f:mathbb R rightarrow mathbb R$ such that for all
              $sum a_n$ CV (resp. ACV), $sum f(a_n)$ is CV (resp. ACV).



              It is a (difficult) exercise to show that for $f:mathbb Rrightarrow mathbb R$,



              $$sum a_n ~CV Rightarrow sum f(a_n) ~CV quad textiff quad exists eta>0,exists lambdain mathbb R,forall xin ]-eta,eta[, quad f(x)=lambda x $$



              $$sum a_n ~CV Rightarrow sum f(a_n) ~ACV quad textiff quad exists eta>0,forall xin ]-eta,eta[, quad f(x)=0 quadquadquad~~$$



              $$sum a_n ~ACV Rightarrow sum f(a_n) ~ACV quad textiff quadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquad$$ $$quad f(0)=0 text and exists eta>0,exists M >0,forall xin ]-eta,eta[, |f(x)|leq M |x| $$



              $$sum a_n ACV Rightarrow sum f(a_n) ~CV quad textiff quad sum a_n ~ACV Rightarrow sum f(a_n) ~ACV~~~~~~~~~~~~~~ $$






              share|cite|improve this answer










              New contributor




              Can I play with Mathness is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
              Check out our Code of Conduct.













              • 1




                $sum_n a_n^2$ converges for this one.
                – Robert Israel
                23 mins ago










              • oups I answer too fast ^^
                – Can I play with Mathness
                20 mins ago










              • Can you correct it? Want to upvote the distinction between ACV and CV but can't bring myself to do it with the error
                – Oscar Lanzi
                11 mins ago










              • corrected, I also added a more general statement.
                – Can I play with Mathness
                10 mins ago














              up vote
              0
              down vote













              The simplest example I have in mind is :
              $$sum_n=1^+inftyfrac(-1)^nsqrtn $$



              Beware that convergent (CV) and absolutely convergent (ACV) can be very different.
              Indeed, if $sum_n=1^+infty a_n $ is ACV then, $sum_n=1^+infty a_n^2$ is also ACV. To prove it, you can simply notice that, since $lim a_n=0$ then for $n$ big enough, $|a_n^2|<|a_n|$.



              You can also ask yourself a more general question :
              What are the function $f:mathbb R rightarrow mathbb R$ such that for all
              $sum a_n$ CV (resp. ACV), $sum f(a_n)$ is CV (resp. ACV).



              It is a (difficult) exercise to show that for $f:mathbb Rrightarrow mathbb R$,



              $$sum a_n ~CV Rightarrow sum f(a_n) ~CV quad textiff quad exists eta>0,exists lambdain mathbb R,forall xin ]-eta,eta[, quad f(x)=lambda x $$



              $$sum a_n ~CV Rightarrow sum f(a_n) ~ACV quad textiff quad exists eta>0,forall xin ]-eta,eta[, quad f(x)=0 quadquadquad~~$$



              $$sum a_n ~ACV Rightarrow sum f(a_n) ~ACV quad textiff quadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquad$$ $$quad f(0)=0 text and exists eta>0,exists M >0,forall xin ]-eta,eta[, |f(x)|leq M |x| $$



              $$sum a_n ACV Rightarrow sum f(a_n) ~CV quad textiff quad sum a_n ~ACV Rightarrow sum f(a_n) ~ACV~~~~~~~~~~~~~~ $$






              share|cite|improve this answer










              New contributor




              Can I play with Mathness is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
              Check out our Code of Conduct.













              • 1




                $sum_n a_n^2$ converges for this one.
                – Robert Israel
                23 mins ago










              • oups I answer too fast ^^
                – Can I play with Mathness
                20 mins ago










              • Can you correct it? Want to upvote the distinction between ACV and CV but can't bring myself to do it with the error
                – Oscar Lanzi
                11 mins ago










              • corrected, I also added a more general statement.
                – Can I play with Mathness
                10 mins ago












              up vote
              0
              down vote










              up vote
              0
              down vote









              The simplest example I have in mind is :
              $$sum_n=1^+inftyfrac(-1)^nsqrtn $$



              Beware that convergent (CV) and absolutely convergent (ACV) can be very different.
              Indeed, if $sum_n=1^+infty a_n $ is ACV then, $sum_n=1^+infty a_n^2$ is also ACV. To prove it, you can simply notice that, since $lim a_n=0$ then for $n$ big enough, $|a_n^2|<|a_n|$.



              You can also ask yourself a more general question :
              What are the function $f:mathbb R rightarrow mathbb R$ such that for all
              $sum a_n$ CV (resp. ACV), $sum f(a_n)$ is CV (resp. ACV).



              It is a (difficult) exercise to show that for $f:mathbb Rrightarrow mathbb R$,



              $$sum a_n ~CV Rightarrow sum f(a_n) ~CV quad textiff quad exists eta>0,exists lambdain mathbb R,forall xin ]-eta,eta[, quad f(x)=lambda x $$



              $$sum a_n ~CV Rightarrow sum f(a_n) ~ACV quad textiff quad exists eta>0,forall xin ]-eta,eta[, quad f(x)=0 quadquadquad~~$$



              $$sum a_n ~ACV Rightarrow sum f(a_n) ~ACV quad textiff quadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquad$$ $$quad f(0)=0 text and exists eta>0,exists M >0,forall xin ]-eta,eta[, |f(x)|leq M |x| $$



              $$sum a_n ACV Rightarrow sum f(a_n) ~CV quad textiff quad sum a_n ~ACV Rightarrow sum f(a_n) ~ACV~~~~~~~~~~~~~~ $$






              share|cite|improve this answer










              New contributor




              Can I play with Mathness is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
              Check out our Code of Conduct.









              The simplest example I have in mind is :
              $$sum_n=1^+inftyfrac(-1)^nsqrtn $$



              Beware that convergent (CV) and absolutely convergent (ACV) can be very different.
              Indeed, if $sum_n=1^+infty a_n $ is ACV then, $sum_n=1^+infty a_n^2$ is also ACV. To prove it, you can simply notice that, since $lim a_n=0$ then for $n$ big enough, $|a_n^2|<|a_n|$.



              You can also ask yourself a more general question :
              What are the function $f:mathbb R rightarrow mathbb R$ such that for all
              $sum a_n$ CV (resp. ACV), $sum f(a_n)$ is CV (resp. ACV).



              It is a (difficult) exercise to show that for $f:mathbb Rrightarrow mathbb R$,



              $$sum a_n ~CV Rightarrow sum f(a_n) ~CV quad textiff quad exists eta>0,exists lambdain mathbb R,forall xin ]-eta,eta[, quad f(x)=lambda x $$



              $$sum a_n ~CV Rightarrow sum f(a_n) ~ACV quad textiff quad exists eta>0,forall xin ]-eta,eta[, quad f(x)=0 quadquadquad~~$$



              $$sum a_n ~ACV Rightarrow sum f(a_n) ~ACV quad textiff quadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquad$$ $$quad f(0)=0 text and exists eta>0,exists M >0,forall xin ]-eta,eta[, |f(x)|leq M |x| $$



              $$sum a_n ACV Rightarrow sum f(a_n) ~CV quad textiff quad sum a_n ~ACV Rightarrow sum f(a_n) ~ACV~~~~~~~~~~~~~~ $$







              share|cite|improve this answer










              New contributor




              Can I play with Mathness is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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              share|cite|improve this answer



              share|cite|improve this answer








              edited 10 mins ago





















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              answered 25 mins ago









              Can I play with Mathness

              1443




              1443




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              • 1




                $sum_n a_n^2$ converges for this one.
                – Robert Israel
                23 mins ago










              • oups I answer too fast ^^
                – Can I play with Mathness
                20 mins ago










              • Can you correct it? Want to upvote the distinction between ACV and CV but can't bring myself to do it with the error
                – Oscar Lanzi
                11 mins ago










              • corrected, I also added a more general statement.
                – Can I play with Mathness
                10 mins ago












              • 1




                $sum_n a_n^2$ converges for this one.
                – Robert Israel
                23 mins ago










              • oups I answer too fast ^^
                – Can I play with Mathness
                20 mins ago










              • Can you correct it? Want to upvote the distinction between ACV and CV but can't bring myself to do it with the error
                – Oscar Lanzi
                11 mins ago










              • corrected, I also added a more general statement.
                – Can I play with Mathness
                10 mins ago







              1




              1




              $sum_n a_n^2$ converges for this one.
              – Robert Israel
              23 mins ago




              $sum_n a_n^2$ converges for this one.
              – Robert Israel
              23 mins ago












              oups I answer too fast ^^
              – Can I play with Mathness
              20 mins ago




              oups I answer too fast ^^
              – Can I play with Mathness
              20 mins ago












              Can you correct it? Want to upvote the distinction between ACV and CV but can't bring myself to do it with the error
              – Oscar Lanzi
              11 mins ago




              Can you correct it? Want to upvote the distinction between ACV and CV but can't bring myself to do it with the error
              – Oscar Lanzi
              11 mins ago












              corrected, I also added a more general statement.
              – Can I play with Mathness
              10 mins ago




              corrected, I also added a more general statement.
              – Can I play with Mathness
              10 mins ago










              up vote
              0
              down vote













              More generally,
              if $a_n = dfrac(-1)^nn^1/(2m)$,
              then
              $sum_n=1^infty a_n
              $

              converges for integer $m ge 0$
              and
              $sum_n=1^infty a_n^2m
              $

              diverges.






              share|cite
























                up vote
                0
                down vote













                More generally,
                if $a_n = dfrac(-1)^nn^1/(2m)$,
                then
                $sum_n=1^infty a_n
                $

                converges for integer $m ge 0$
                and
                $sum_n=1^infty a_n^2m
                $

                diverges.






                share|cite






















                  up vote
                  0
                  down vote










                  up vote
                  0
                  down vote









                  More generally,
                  if $a_n = dfrac(-1)^nn^1/(2m)$,
                  then
                  $sum_n=1^infty a_n
                  $

                  converges for integer $m ge 0$
                  and
                  $sum_n=1^infty a_n^2m
                  $

                  diverges.






                  share|cite












                  More generally,
                  if $a_n = dfrac(-1)^nn^1/(2m)$,
                  then
                  $sum_n=1^infty a_n
                  $

                  converges for integer $m ge 0$
                  and
                  $sum_n=1^infty a_n^2m
                  $

                  diverges.







                  share|cite












                  share|cite



                  share|cite










                  answered 1 min ago









                  marty cohen

                  70.9k546122




                  70.9k546122




















                      Snop D. is a new contributor. Be nice, and check out our Code of Conduct.









                       

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