Is there a comprehensive list of convergence tests for double series?












3














I’m looking for a comprehensive list of methods to check for convergence of infinite double series. Specifically conditional convergence.






sequences-and-series convergence







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    3














    I’m looking for a comprehensive list of methods to check for convergence of infinite double series. Specifically conditional convergence.






    sequences-and-series convergence







    share|cite|improve this question















    This question has an open bounty worth +100
    reputation from Sudeepan Datta ending in 6 days.


    This question has not received enough attention.


    The existing response to this question is far from being comprehensive. I think it would be beneficial for all, if a really sincere, detailed answer, carefully explaining the current state-of-affairs as far as this question is concerned, could be obtained from this platform.



















      3












      3








      3


      1





      I’m looking for a comprehensive list of methods to check for convergence of infinite double series. Specifically conditional convergence.






      sequences-and-series convergence







      share|cite|improve this question













      I’m looking for a comprehensive list of methods to check for convergence of infinite double series. Specifically conditional convergence.






      sequences-and-series convergence




      sequences-and-series convergence






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Sep 30 '18 at 19:26









      MathPonderer

      263




      263






      This question has an open bounty worth +100
      reputation from Sudeepan Datta ending in 6 days.


      This question has not received enough attention.


      The existing response to this question is far from being comprehensive. I think it would be beneficial for all, if a really sincere, detailed answer, carefully explaining the current state-of-affairs as far as this question is concerned, could be obtained from this platform.








      This question has an open bounty worth +100
      reputation from Sudeepan Datta ending in 6 days.


      This question has not received enough attention.


      The existing response to this question is far from being comprehensive. I think it would be beneficial for all, if a really sincere, detailed answer, carefully explaining the current state-of-affairs as far as this question is concerned, could be obtained from this platform.
























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          First of all, you have to be careful with exactly what you mean by "infinite double series", because rearrangements of conditionally convergent series can diverge. I'll assume that when you talk about
          convergence of a double series of the form



          $$ sum_{i=1}^infty sum_{j=1}^infty a_{ij}$$
          you mean that for each $i$, the series $sum_{j=1}^infty a_{ij}$ converges to a value $b_i$, and the series $sum_{i=1}^infty b_i$ converges.



          The methods for proving convergence of such double series are essentially the same as those for a single series, except that it may be more difficult because you might not know $b_i$ in closed form.






          share|cite|improve this answer





















          • Does it get any easier if you assume absolute convergence?
            – Theo Bendit
            Sep 30 '18 at 20:12










          • @TheoBendit, if you go looking for absolute convergence (e.g. when the terms in the sum are all positive, so that convergence is equivalent to absolute convergence) then you can start rearranging the sum in interesting ways. For example, it might be that $sum_{k = 2}^infty sum_{i = 1}^{k-1} a_{i,k-i}$ is easier to evaluate, because the finite summation has a nice closed form.
            – Mees de Vries
            13 hours ago













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          1 Answer
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          active

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          0














          First of all, you have to be careful with exactly what you mean by "infinite double series", because rearrangements of conditionally convergent series can diverge. I'll assume that when you talk about
          convergence of a double series of the form



          $$ sum_{i=1}^infty sum_{j=1}^infty a_{ij}$$
          you mean that for each $i$, the series $sum_{j=1}^infty a_{ij}$ converges to a value $b_i$, and the series $sum_{i=1}^infty b_i$ converges.



          The methods for proving convergence of such double series are essentially the same as those for a single series, except that it may be more difficult because you might not know $b_i$ in closed form.






          share|cite|improve this answer





















          • Does it get any easier if you assume absolute convergence?
            – Theo Bendit
            Sep 30 '18 at 20:12










          • @TheoBendit, if you go looking for absolute convergence (e.g. when the terms in the sum are all positive, so that convergence is equivalent to absolute convergence) then you can start rearranging the sum in interesting ways. For example, it might be that $sum_{k = 2}^infty sum_{i = 1}^{k-1} a_{i,k-i}$ is easier to evaluate, because the finite summation has a nice closed form.
            – Mees de Vries
            13 hours ago


















          0














          First of all, you have to be careful with exactly what you mean by "infinite double series", because rearrangements of conditionally convergent series can diverge. I'll assume that when you talk about
          convergence of a double series of the form



          $$ sum_{i=1}^infty sum_{j=1}^infty a_{ij}$$
          you mean that for each $i$, the series $sum_{j=1}^infty a_{ij}$ converges to a value $b_i$, and the series $sum_{i=1}^infty b_i$ converges.



          The methods for proving convergence of such double series are essentially the same as those for a single series, except that it may be more difficult because you might not know $b_i$ in closed form.






          share|cite|improve this answer





















          • Does it get any easier if you assume absolute convergence?
            – Theo Bendit
            Sep 30 '18 at 20:12










          • @TheoBendit, if you go looking for absolute convergence (e.g. when the terms in the sum are all positive, so that convergence is equivalent to absolute convergence) then you can start rearranging the sum in interesting ways. For example, it might be that $sum_{k = 2}^infty sum_{i = 1}^{k-1} a_{i,k-i}$ is easier to evaluate, because the finite summation has a nice closed form.
            – Mees de Vries
            13 hours ago
















          0












          0








          0






          First of all, you have to be careful with exactly what you mean by "infinite double series", because rearrangements of conditionally convergent series can diverge. I'll assume that when you talk about
          convergence of a double series of the form



          $$ sum_{i=1}^infty sum_{j=1}^infty a_{ij}$$
          you mean that for each $i$, the series $sum_{j=1}^infty a_{ij}$ converges to a value $b_i$, and the series $sum_{i=1}^infty b_i$ converges.



          The methods for proving convergence of such double series are essentially the same as those for a single series, except that it may be more difficult because you might not know $b_i$ in closed form.






          share|cite|improve this answer












          First of all, you have to be careful with exactly what you mean by "infinite double series", because rearrangements of conditionally convergent series can diverge. I'll assume that when you talk about
          convergence of a double series of the form



          $$ sum_{i=1}^infty sum_{j=1}^infty a_{ij}$$
          you mean that for each $i$, the series $sum_{j=1}^infty a_{ij}$ converges to a value $b_i$, and the series $sum_{i=1}^infty b_i$ converges.



          The methods for proving convergence of such double series are essentially the same as those for a single series, except that it may be more difficult because you might not know $b_i$ in closed form.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Sep 30 '18 at 20:05









          Robert Israel

          318k23208457




          318k23208457












          • Does it get any easier if you assume absolute convergence?
            – Theo Bendit
            Sep 30 '18 at 20:12










          • @TheoBendit, if you go looking for absolute convergence (e.g. when the terms in the sum are all positive, so that convergence is equivalent to absolute convergence) then you can start rearranging the sum in interesting ways. For example, it might be that $sum_{k = 2}^infty sum_{i = 1}^{k-1} a_{i,k-i}$ is easier to evaluate, because the finite summation has a nice closed form.
            – Mees de Vries
            13 hours ago




















          • Does it get any easier if you assume absolute convergence?
            – Theo Bendit
            Sep 30 '18 at 20:12










          • @TheoBendit, if you go looking for absolute convergence (e.g. when the terms in the sum are all positive, so that convergence is equivalent to absolute convergence) then you can start rearranging the sum in interesting ways. For example, it might be that $sum_{k = 2}^infty sum_{i = 1}^{k-1} a_{i,k-i}$ is easier to evaluate, because the finite summation has a nice closed form.
            – Mees de Vries
            13 hours ago


















          Does it get any easier if you assume absolute convergence?
          – Theo Bendit
          Sep 30 '18 at 20:12




          Does it get any easier if you assume absolute convergence?
          – Theo Bendit
          Sep 30 '18 at 20:12












          @TheoBendit, if you go looking for absolute convergence (e.g. when the terms in the sum are all positive, so that convergence is equivalent to absolute convergence) then you can start rearranging the sum in interesting ways. For example, it might be that $sum_{k = 2}^infty sum_{i = 1}^{k-1} a_{i,k-i}$ is easier to evaluate, because the finite summation has a nice closed form.
          – Mees de Vries
          13 hours ago






          @TheoBendit, if you go looking for absolute convergence (e.g. when the terms in the sum are all positive, so that convergence is equivalent to absolute convergence) then you can start rearranging the sum in interesting ways. For example, it might be that $sum_{k = 2}^infty sum_{i = 1}^{k-1} a_{i,k-i}$ is easier to evaluate, because the finite summation has a nice closed form.
          – Mees de Vries
          13 hours ago




















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