Difference between divergent series and series with no limit?












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Can a series have a limit and be divergent? I'm confused about the difference between divergence and a series not having a limit. A ck-12 calculus book stated they are different concepts.






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    The limit could be $infty$ for example, in which case we still say it's divergent.
    $endgroup$
    – Najib Idrissi
    Jun 18 '13 at 19:04










  • $begingroup$
    If you are reading this from a calculus book, make sure you are talking about the same objects having a limit. For example, a series $sum_{n = 1}^infty a_n$ converges if and only if the sequence of partial sums $s_k = sum_{n = 1}^k a_n$ has a limit as $k$ approaches $infty$. This is not the same as determining if the sequence $a_n$ has a limit as $n to infty$.
    $endgroup$
    – Ben Passer
    Jun 18 '13 at 21:09


















1












$begingroup$


Can a series have a limit and be divergent? I'm confused about the difference between divergence and a series not having a limit. A ck-12 calculus book stated they are different concepts.






sequences-and-series







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    The limit could be $infty$ for example, in which case we still say it's divergent.
    $endgroup$
    – Najib Idrissi
    Jun 18 '13 at 19:04










  • $begingroup$
    If you are reading this from a calculus book, make sure you are talking about the same objects having a limit. For example, a series $sum_{n = 1}^infty a_n$ converges if and only if the sequence of partial sums $s_k = sum_{n = 1}^k a_n$ has a limit as $k$ approaches $infty$. This is not the same as determining if the sequence $a_n$ has a limit as $n to infty$.
    $endgroup$
    – Ben Passer
    Jun 18 '13 at 21:09
















1












1








1





$begingroup$


Can a series have a limit and be divergent? I'm confused about the difference between divergence and a series not having a limit. A ck-12 calculus book stated they are different concepts.






sequences-and-series







share|cite|improve this question









$endgroup$




Can a series have a limit and be divergent? I'm confused about the difference between divergence and a series not having a limit. A ck-12 calculus book stated they are different concepts.






sequences-and-series




sequences-and-series






share|cite|improve this question













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asked Jun 18 '13 at 19:02









Aaron BooherAaron Booher

814




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  • 1




    $begingroup$
    The limit could be $infty$ for example, in which case we still say it's divergent.
    $endgroup$
    – Najib Idrissi
    Jun 18 '13 at 19:04










  • $begingroup$
    If you are reading this from a calculus book, make sure you are talking about the same objects having a limit. For example, a series $sum_{n = 1}^infty a_n$ converges if and only if the sequence of partial sums $s_k = sum_{n = 1}^k a_n$ has a limit as $k$ approaches $infty$. This is not the same as determining if the sequence $a_n$ has a limit as $n to infty$.
    $endgroup$
    – Ben Passer
    Jun 18 '13 at 21:09
















  • 1




    $begingroup$
    The limit could be $infty$ for example, in which case we still say it's divergent.
    $endgroup$
    – Najib Idrissi
    Jun 18 '13 at 19:04










  • $begingroup$
    If you are reading this from a calculus book, make sure you are talking about the same objects having a limit. For example, a series $sum_{n = 1}^infty a_n$ converges if and only if the sequence of partial sums $s_k = sum_{n = 1}^k a_n$ has a limit as $k$ approaches $infty$. This is not the same as determining if the sequence $a_n$ has a limit as $n to infty$.
    $endgroup$
    – Ben Passer
    Jun 18 '13 at 21:09










1




1




$begingroup$
The limit could be $infty$ for example, in which case we still say it's divergent.
$endgroup$
– Najib Idrissi
Jun 18 '13 at 19:04




$begingroup$
The limit could be $infty$ for example, in which case we still say it's divergent.
$endgroup$
– Najib Idrissi
Jun 18 '13 at 19:04












$begingroup$
If you are reading this from a calculus book, make sure you are talking about the same objects having a limit. For example, a series $sum_{n = 1}^infty a_n$ converges if and only if the sequence of partial sums $s_k = sum_{n = 1}^k a_n$ has a limit as $k$ approaches $infty$. This is not the same as determining if the sequence $a_n$ has a limit as $n to infty$.
$endgroup$
– Ben Passer
Jun 18 '13 at 21:09






$begingroup$
If you are reading this from a calculus book, make sure you are talking about the same objects having a limit. For example, a series $sum_{n = 1}^infty a_n$ converges if and only if the sequence of partial sums $s_k = sum_{n = 1}^k a_n$ has a limit as $k$ approaches $infty$. This is not the same as determining if the sequence $a_n$ has a limit as $n to infty$.
$endgroup$
– Ben Passer
Jun 18 '13 at 21:09












2 Answers
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If a series is convergent then it has bounded partial sums. Equivalently, if a series fails to have bounded partial sums then it is divergent.



The converse is false. If a series has bounded partial sums then it need not be convergent. A counter-example is $sum_{n ge 1}(-1)^n$ where the partial sums are $-1,0,-1,0,-1,ldots$.



The partial sums obtain a finite limit if and only if the series is convergent.






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$endgroup$













  • $begingroup$
    Boundedness of partial sums is a sufficient condition if the terms are nonnegative, though.
    $endgroup$
    – Math1000
    May 27 '15 at 13:57



















0












$begingroup$

In every calculus and real analysis book I've read, the two are the same thing.For example, in Stewart's calculus:





  1. A sequence ${a_n}$ has the limit $L$ and we write $$lim_{ntoinfty}a_n=Lquadtext{or}quad a_nto Ltext{ as }ntoinfty$$ if we can make the terms $a_n$ as close to $L$ as we like by taking $n$ sufficiently large. If $lim_{ntoinfty}$ exists, we say the sequence converges (or is convergent). Otherwise we say the sequence diverges (or is divergent).


  2. Given a series $sum_{n=1}^infty a_n=a_1+a_2+a_3+cdots$, let $s_n$ denote its $n$th partial sum: $$s_n=sum_{i=1}^na_i=a_1+a_2+cdots+a_n$$ If the sequence ${s_n}$ is convergent and $lim_{ntoinfty}s_n=s$ exists as a real number, then the series $Sigma a_n$ is called convergent and we write $$a_1+a_2+cdots+a_n+cdots=squadtext{or}quadsum_{n=1}^infty a_n=s$$ The number $s$ is called the sum of the series. If the sequence ${s_n}$ is divergent, then the series is called divergent.





Either a series converges, or it diverges. Converge means it has a limit. Diverge means it doesn't.






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    2 Answers
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    2 Answers
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    active

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    0












    $begingroup$

    If a series is convergent then it has bounded partial sums. Equivalently, if a series fails to have bounded partial sums then it is divergent.



    The converse is false. If a series has bounded partial sums then it need not be convergent. A counter-example is $sum_{n ge 1}(-1)^n$ where the partial sums are $-1,0,-1,0,-1,ldots$.



    The partial sums obtain a finite limit if and only if the series is convergent.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      Boundedness of partial sums is a sufficient condition if the terms are nonnegative, though.
      $endgroup$
      – Math1000
      May 27 '15 at 13:57
















    0












    $begingroup$

    If a series is convergent then it has bounded partial sums. Equivalently, if a series fails to have bounded partial sums then it is divergent.



    The converse is false. If a series has bounded partial sums then it need not be convergent. A counter-example is $sum_{n ge 1}(-1)^n$ where the partial sums are $-1,0,-1,0,-1,ldots$.



    The partial sums obtain a finite limit if and only if the series is convergent.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      Boundedness of partial sums is a sufficient condition if the terms are nonnegative, though.
      $endgroup$
      – Math1000
      May 27 '15 at 13:57














    0












    0








    0





    $begingroup$

    If a series is convergent then it has bounded partial sums. Equivalently, if a series fails to have bounded partial sums then it is divergent.



    The converse is false. If a series has bounded partial sums then it need not be convergent. A counter-example is $sum_{n ge 1}(-1)^n$ where the partial sums are $-1,0,-1,0,-1,ldots$.



    The partial sums obtain a finite limit if and only if the series is convergent.






    share|cite|improve this answer









    $endgroup$



    If a series is convergent then it has bounded partial sums. Equivalently, if a series fails to have bounded partial sums then it is divergent.



    The converse is false. If a series has bounded partial sums then it need not be convergent. A counter-example is $sum_{n ge 1}(-1)^n$ where the partial sums are $-1,0,-1,0,-1,ldots$.



    The partial sums obtain a finite limit if and only if the series is convergent.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Jun 18 '13 at 19:20









    Fly by NightFly by Night

    25.9k32978




    25.9k32978












    • $begingroup$
      Boundedness of partial sums is a sufficient condition if the terms are nonnegative, though.
      $endgroup$
      – Math1000
      May 27 '15 at 13:57


















    • $begingroup$
      Boundedness of partial sums is a sufficient condition if the terms are nonnegative, though.
      $endgroup$
      – Math1000
      May 27 '15 at 13:57
















    $begingroup$
    Boundedness of partial sums is a sufficient condition if the terms are nonnegative, though.
    $endgroup$
    – Math1000
    May 27 '15 at 13:57




    $begingroup$
    Boundedness of partial sums is a sufficient condition if the terms are nonnegative, though.
    $endgroup$
    – Math1000
    May 27 '15 at 13:57











    0












    $begingroup$

    In every calculus and real analysis book I've read, the two are the same thing.For example, in Stewart's calculus:





    1. A sequence ${a_n}$ has the limit $L$ and we write $$lim_{ntoinfty}a_n=Lquadtext{or}quad a_nto Ltext{ as }ntoinfty$$ if we can make the terms $a_n$ as close to $L$ as we like by taking $n$ sufficiently large. If $lim_{ntoinfty}$ exists, we say the sequence converges (or is convergent). Otherwise we say the sequence diverges (or is divergent).


    2. Given a series $sum_{n=1}^infty a_n=a_1+a_2+a_3+cdots$, let $s_n$ denote its $n$th partial sum: $$s_n=sum_{i=1}^na_i=a_1+a_2+cdots+a_n$$ If the sequence ${s_n}$ is convergent and $lim_{ntoinfty}s_n=s$ exists as a real number, then the series $Sigma a_n$ is called convergent and we write $$a_1+a_2+cdots+a_n+cdots=squadtext{or}quadsum_{n=1}^infty a_n=s$$ The number $s$ is called the sum of the series. If the sequence ${s_n}$ is divergent, then the series is called divergent.





    Either a series converges, or it diverges. Converge means it has a limit. Diverge means it doesn't.






    share|cite|improve this answer











    $endgroup$


















      0












      $begingroup$

      In every calculus and real analysis book I've read, the two are the same thing.For example, in Stewart's calculus:





      1. A sequence ${a_n}$ has the limit $L$ and we write $$lim_{ntoinfty}a_n=Lquadtext{or}quad a_nto Ltext{ as }ntoinfty$$ if we can make the terms $a_n$ as close to $L$ as we like by taking $n$ sufficiently large. If $lim_{ntoinfty}$ exists, we say the sequence converges (or is convergent). Otherwise we say the sequence diverges (or is divergent).


      2. Given a series $sum_{n=1}^infty a_n=a_1+a_2+a_3+cdots$, let $s_n$ denote its $n$th partial sum: $$s_n=sum_{i=1}^na_i=a_1+a_2+cdots+a_n$$ If the sequence ${s_n}$ is convergent and $lim_{ntoinfty}s_n=s$ exists as a real number, then the series $Sigma a_n$ is called convergent and we write $$a_1+a_2+cdots+a_n+cdots=squadtext{or}quadsum_{n=1}^infty a_n=s$$ The number $s$ is called the sum of the series. If the sequence ${s_n}$ is divergent, then the series is called divergent.





      Either a series converges, or it diverges. Converge means it has a limit. Diverge means it doesn't.






      share|cite|improve this answer











      $endgroup$
















        0












        0








        0





        $begingroup$

        In every calculus and real analysis book I've read, the two are the same thing.For example, in Stewart's calculus:





        1. A sequence ${a_n}$ has the limit $L$ and we write $$lim_{ntoinfty}a_n=Lquadtext{or}quad a_nto Ltext{ as }ntoinfty$$ if we can make the terms $a_n$ as close to $L$ as we like by taking $n$ sufficiently large. If $lim_{ntoinfty}$ exists, we say the sequence converges (or is convergent). Otherwise we say the sequence diverges (or is divergent).


        2. Given a series $sum_{n=1}^infty a_n=a_1+a_2+a_3+cdots$, let $s_n$ denote its $n$th partial sum: $$s_n=sum_{i=1}^na_i=a_1+a_2+cdots+a_n$$ If the sequence ${s_n}$ is convergent and $lim_{ntoinfty}s_n=s$ exists as a real number, then the series $Sigma a_n$ is called convergent and we write $$a_1+a_2+cdots+a_n+cdots=squadtext{or}quadsum_{n=1}^infty a_n=s$$ The number $s$ is called the sum of the series. If the sequence ${s_n}$ is divergent, then the series is called divergent.





        Either a series converges, or it diverges. Converge means it has a limit. Diverge means it doesn't.






        share|cite|improve this answer











        $endgroup$



        In every calculus and real analysis book I've read, the two are the same thing.For example, in Stewart's calculus:





        1. A sequence ${a_n}$ has the limit $L$ and we write $$lim_{ntoinfty}a_n=Lquadtext{or}quad a_nto Ltext{ as }ntoinfty$$ if we can make the terms $a_n$ as close to $L$ as we like by taking $n$ sufficiently large. If $lim_{ntoinfty}$ exists, we say the sequence converges (or is convergent). Otherwise we say the sequence diverges (or is divergent).


        2. Given a series $sum_{n=1}^infty a_n=a_1+a_2+a_3+cdots$, let $s_n$ denote its $n$th partial sum: $$s_n=sum_{i=1}^na_i=a_1+a_2+cdots+a_n$$ If the sequence ${s_n}$ is convergent and $lim_{ntoinfty}s_n=s$ exists as a real number, then the series $Sigma a_n$ is called convergent and we write $$a_1+a_2+cdots+a_n+cdots=squadtext{or}quadsum_{n=1}^infty a_n=s$$ The number $s$ is called the sum of the series. If the sequence ${s_n}$ is divergent, then the series is called divergent.





        Either a series converges, or it diverges. Converge means it has a limit. Diverge means it doesn't.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Jan 7 '17 at 15:04









        grg

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        answered Apr 25 '15 at 7:56







        user223391





































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