What is the definition of $sum_{n = - infty}^{infty} a_n$












0












$begingroup$


I know that in Complex Analysis we use sums of the form $displaystylesum_{n = - infty}^{infty} a_n$



What is the actual meaning of this symbol? I expect that under some nice enough conditions, it would be equal to something like $displaystyle sum_{n=1}^infty b_n$, where $b_{2k} = a_k$ and $b_{2k+1} = a_{-k}$. But what is the actual definition?






complex-analysis analysis laurent-series







share|cite|improve this question









$endgroup$

















    0












    $begingroup$


    I know that in Complex Analysis we use sums of the form $displaystylesum_{n = - infty}^{infty} a_n$



    What is the actual meaning of this symbol? I expect that under some nice enough conditions, it would be equal to something like $displaystyle sum_{n=1}^infty b_n$, where $b_{2k} = a_k$ and $b_{2k+1} = a_{-k}$. But what is the actual definition?






    complex-analysis analysis laurent-series







    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      I know that in Complex Analysis we use sums of the form $displaystylesum_{n = - infty}^{infty} a_n$



      What is the actual meaning of this symbol? I expect that under some nice enough conditions, it would be equal to something like $displaystyle sum_{n=1}^infty b_n$, where $b_{2k} = a_k$ and $b_{2k+1} = a_{-k}$. But what is the actual definition?






      complex-analysis analysis laurent-series







      share|cite|improve this question









      $endgroup$




      I know that in Complex Analysis we use sums of the form $displaystylesum_{n = - infty}^{infty} a_n$



      What is the actual meaning of this symbol? I expect that under some nice enough conditions, it would be equal to something like $displaystyle sum_{n=1}^infty b_n$, where $b_{2k} = a_k$ and $b_{2k+1} = a_{-k}$. But what is the actual definition?






      complex-analysis analysis laurent-series




      complex-analysis analysis laurent-series






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 3 '18 at 1:22









      OviOvi

      12.4k1038112




      12.4k1038112






















          1 Answer
          1






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          3












          $begingroup$

          The sum is defined as $$displaystylesum_{n = - infty}^{infty} a_n:= lim_{Ntoinfty}sum_{n=0}^{N} a_n+lim_{Mtoinfty}sum_{n=1}^{M} a_{-n}$$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            In particular, we need both the positive subsequence and the negative subsequence to exist, instead of picking some Cauchy principal value.
            $endgroup$
            – obscurans
            Dec 3 '18 at 1:41










          • $begingroup$
            Thanks for the answer. Are there any theorems which allow you to play around with the sums, such as I did in my post?
            $endgroup$
            – Ovi
            Dec 3 '18 at 1:42










          • $begingroup$
            The usual stuff with rearranging two infinite sums - absolute convergence allows you to reorder terms.
            $endgroup$
            – obscurans
            Dec 3 '18 at 1:52










          • $begingroup$
            If you know that the sums are absolutely convergent (i.e. $sum|a_n|$ converges), then you may rearrange terms and sum in whatever manner you like, including defining the sequence $b_n$ as you did. However, if the sum is conditionally convergent (i.e. convergent but not absolutely convergent), then the Riemann series theorem states that you can rearrange the terms of the series so that the sum converges to any number you like, or diverges.
            $endgroup$
            – greelious
            Dec 3 '18 at 1:54











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          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          3












          $begingroup$

          The sum is defined as $$displaystylesum_{n = - infty}^{infty} a_n:= lim_{Ntoinfty}sum_{n=0}^{N} a_n+lim_{Mtoinfty}sum_{n=1}^{M} a_{-n}$$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            In particular, we need both the positive subsequence and the negative subsequence to exist, instead of picking some Cauchy principal value.
            $endgroup$
            – obscurans
            Dec 3 '18 at 1:41










          • $begingroup$
            Thanks for the answer. Are there any theorems which allow you to play around with the sums, such as I did in my post?
            $endgroup$
            – Ovi
            Dec 3 '18 at 1:42










          • $begingroup$
            The usual stuff with rearranging two infinite sums - absolute convergence allows you to reorder terms.
            $endgroup$
            – obscurans
            Dec 3 '18 at 1:52










          • $begingroup$
            If you know that the sums are absolutely convergent (i.e. $sum|a_n|$ converges), then you may rearrange terms and sum in whatever manner you like, including defining the sequence $b_n$ as you did. However, if the sum is conditionally convergent (i.e. convergent but not absolutely convergent), then the Riemann series theorem states that you can rearrange the terms of the series so that the sum converges to any number you like, or diverges.
            $endgroup$
            – greelious
            Dec 3 '18 at 1:54
















          3












          $begingroup$

          The sum is defined as $$displaystylesum_{n = - infty}^{infty} a_n:= lim_{Ntoinfty}sum_{n=0}^{N} a_n+lim_{Mtoinfty}sum_{n=1}^{M} a_{-n}$$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            In particular, we need both the positive subsequence and the negative subsequence to exist, instead of picking some Cauchy principal value.
            $endgroup$
            – obscurans
            Dec 3 '18 at 1:41










          • $begingroup$
            Thanks for the answer. Are there any theorems which allow you to play around with the sums, such as I did in my post?
            $endgroup$
            – Ovi
            Dec 3 '18 at 1:42










          • $begingroup$
            The usual stuff with rearranging two infinite sums - absolute convergence allows you to reorder terms.
            $endgroup$
            – obscurans
            Dec 3 '18 at 1:52










          • $begingroup$
            If you know that the sums are absolutely convergent (i.e. $sum|a_n|$ converges), then you may rearrange terms and sum in whatever manner you like, including defining the sequence $b_n$ as you did. However, if the sum is conditionally convergent (i.e. convergent but not absolutely convergent), then the Riemann series theorem states that you can rearrange the terms of the series so that the sum converges to any number you like, or diverges.
            $endgroup$
            – greelious
            Dec 3 '18 at 1:54














          3












          3








          3





          $begingroup$

          The sum is defined as $$displaystylesum_{n = - infty}^{infty} a_n:= lim_{Ntoinfty}sum_{n=0}^{N} a_n+lim_{Mtoinfty}sum_{n=1}^{M} a_{-n}$$






          share|cite|improve this answer









          $endgroup$



          The sum is defined as $$displaystylesum_{n = - infty}^{infty} a_n:= lim_{Ntoinfty}sum_{n=0}^{N} a_n+lim_{Mtoinfty}sum_{n=1}^{M} a_{-n}$$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 3 '18 at 1:26









          greeliousgreelious

          19410




          19410












          • $begingroup$
            In particular, we need both the positive subsequence and the negative subsequence to exist, instead of picking some Cauchy principal value.
            $endgroup$
            – obscurans
            Dec 3 '18 at 1:41










          • $begingroup$
            Thanks for the answer. Are there any theorems which allow you to play around with the sums, such as I did in my post?
            $endgroup$
            – Ovi
            Dec 3 '18 at 1:42










          • $begingroup$
            The usual stuff with rearranging two infinite sums - absolute convergence allows you to reorder terms.
            $endgroup$
            – obscurans
            Dec 3 '18 at 1:52










          • $begingroup$
            If you know that the sums are absolutely convergent (i.e. $sum|a_n|$ converges), then you may rearrange terms and sum in whatever manner you like, including defining the sequence $b_n$ as you did. However, if the sum is conditionally convergent (i.e. convergent but not absolutely convergent), then the Riemann series theorem states that you can rearrange the terms of the series so that the sum converges to any number you like, or diverges.
            $endgroup$
            – greelious
            Dec 3 '18 at 1:54


















          • $begingroup$
            In particular, we need both the positive subsequence and the negative subsequence to exist, instead of picking some Cauchy principal value.
            $endgroup$
            – obscurans
            Dec 3 '18 at 1:41










          • $begingroup$
            Thanks for the answer. Are there any theorems which allow you to play around with the sums, such as I did in my post?
            $endgroup$
            – Ovi
            Dec 3 '18 at 1:42










          • $begingroup$
            The usual stuff with rearranging two infinite sums - absolute convergence allows you to reorder terms.
            $endgroup$
            – obscurans
            Dec 3 '18 at 1:52










          • $begingroup$
            If you know that the sums are absolutely convergent (i.e. $sum|a_n|$ converges), then you may rearrange terms and sum in whatever manner you like, including defining the sequence $b_n$ as you did. However, if the sum is conditionally convergent (i.e. convergent but not absolutely convergent), then the Riemann series theorem states that you can rearrange the terms of the series so that the sum converges to any number you like, or diverges.
            $endgroup$
            – greelious
            Dec 3 '18 at 1:54
















          $begingroup$
          In particular, we need both the positive subsequence and the negative subsequence to exist, instead of picking some Cauchy principal value.
          $endgroup$
          – obscurans
          Dec 3 '18 at 1:41




          $begingroup$
          In particular, we need both the positive subsequence and the negative subsequence to exist, instead of picking some Cauchy principal value.
          $endgroup$
          – obscurans
          Dec 3 '18 at 1:41












          $begingroup$
          Thanks for the answer. Are there any theorems which allow you to play around with the sums, such as I did in my post?
          $endgroup$
          – Ovi
          Dec 3 '18 at 1:42




          $begingroup$
          Thanks for the answer. Are there any theorems which allow you to play around with the sums, such as I did in my post?
          $endgroup$
          – Ovi
          Dec 3 '18 at 1:42












          $begingroup$
          The usual stuff with rearranging two infinite sums - absolute convergence allows you to reorder terms.
          $endgroup$
          – obscurans
          Dec 3 '18 at 1:52




          $begingroup$
          The usual stuff with rearranging two infinite sums - absolute convergence allows you to reorder terms.
          $endgroup$
          – obscurans
          Dec 3 '18 at 1:52












          $begingroup$
          If you know that the sums are absolutely convergent (i.e. $sum|a_n|$ converges), then you may rearrange terms and sum in whatever manner you like, including defining the sequence $b_n$ as you did. However, if the sum is conditionally convergent (i.e. convergent but not absolutely convergent), then the Riemann series theorem states that you can rearrange the terms of the series so that the sum converges to any number you like, or diverges.
          $endgroup$
          – greelious
          Dec 3 '18 at 1:54




          $begingroup$
          If you know that the sums are absolutely convergent (i.e. $sum|a_n|$ converges), then you may rearrange terms and sum in whatever manner you like, including defining the sequence $b_n$ as you did. However, if the sum is conditionally convergent (i.e. convergent but not absolutely convergent), then the Riemann series theorem states that you can rearrange the terms of the series so that the sum converges to any number you like, or diverges.
          $endgroup$
          – greelious
          Dec 3 '18 at 1:54


















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