Convergence of term-wise product of convergent series











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Let $sum a_n$ and $sum b_n$ be two convergent series. It is easy to prove that their term-wise product $sum a_n b_n$ converges if $a_n,b_n geq 0$, but $sum a_n b_n$ does not necessarily converge otherwise.



My question is, must $sum a_n b_n$ converge if $a_n geq 0$? Having thought about it some, it seems that there should be a counterexample, but I haven't been able to find one.






calculus real-analysis sequences-and-series limits convergence







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    Let $sum a_n$ and $sum b_n$ be two convergent series. It is easy to prove that their term-wise product $sum a_n b_n$ converges if $a_n,b_n geq 0$, but $sum a_n b_n$ does not necessarily converge otherwise.



    My question is, must $sum a_n b_n$ converge if $a_n geq 0$? Having thought about it some, it seems that there should be a counterexample, but I haven't been able to find one.






    calculus real-analysis sequences-and-series limits convergence







    share|cite|improve this question







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    AlephNull is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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      up vote
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      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Let $sum a_n$ and $sum b_n$ be two convergent series. It is easy to prove that their term-wise product $sum a_n b_n$ converges if $a_n,b_n geq 0$, but $sum a_n b_n$ does not necessarily converge otherwise.



      My question is, must $sum a_n b_n$ converge if $a_n geq 0$? Having thought about it some, it seems that there should be a counterexample, but I haven't been able to find one.






      calculus real-analysis sequences-and-series limits convergence







      share|cite|improve this question







      New contributor




      AlephNull is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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      Let $sum a_n$ and $sum b_n$ be two convergent series. It is easy to prove that their term-wise product $sum a_n b_n$ converges if $a_n,b_n geq 0$, but $sum a_n b_n$ does not necessarily converge otherwise.



      My question is, must $sum a_n b_n$ converge if $a_n geq 0$? Having thought about it some, it seems that there should be a counterexample, but I haven't been able to find one.






      calculus real-analysis sequences-and-series limits convergence




      calculus real-analysis sequences-and-series limits convergence






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          $sum |a_n b_n| leq Msum a_n < infty$ where $M=sup_n |b_n|$. Note that $b_n to 0$ so ${b_n}$ is a bounded sequence. Hence $M <infty$ and the series $sum a_n b_n$ is absolutely convergent.






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            $sum |a_n b_n| leq Msum a_n < infty$ where $M=sup_n |b_n|$. Note that $b_n to 0$ so ${b_n}$ is a bounded sequence. Hence $M <infty$ and the series $sum a_n b_n$ is absolutely convergent.






            share|cite|improve this answer

























              up vote
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              accepted










              $sum |a_n b_n| leq Msum a_n < infty$ where $M=sup_n |b_n|$. Note that $b_n to 0$ so ${b_n}$ is a bounded sequence. Hence $M <infty$ and the series $sum a_n b_n$ is absolutely convergent.






              share|cite|improve this answer























                up vote
                1
                down vote



                accepted







                up vote
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                accepted






                $sum |a_n b_n| leq Msum a_n < infty$ where $M=sup_n |b_n|$. Note that $b_n to 0$ so ${b_n}$ is a bounded sequence. Hence $M <infty$ and the series $sum a_n b_n$ is absolutely convergent.






                share|cite|improve this answer












                $sum |a_n b_n| leq Msum a_n < infty$ where $M=sup_n |b_n|$. Note that $b_n to 0$ so ${b_n}$ is a bounded sequence. Hence $M <infty$ and the series $sum a_n b_n$ is absolutely convergent.







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                share|cite|improve this answer










                answered 2 days ago









                Kavi Rama Murthy

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