A (probably easy) question about exchanging the order of this double sum












0












$begingroup$


I would like to exchange the order of the following:



$sum_{k=1}^{i-1}sum_{r=1}^{2k}$ (stuff).



I feel like it should be easy, but so far I am only able to produce



$sum_{r=1}^{2(i-1)}sum_{k=??}^{??}$ (stuff).



In particular, I don't see how to reconcile the inequalities $1 leq rleq 2k,$ and $1 leq k leq i-1$, without introducing half integers into the sum.



Thanks,






summation







share|cite|improve this question









$endgroup$

















    0












    $begingroup$


    I would like to exchange the order of the following:



    $sum_{k=1}^{i-1}sum_{r=1}^{2k}$ (stuff).



    I feel like it should be easy, but so far I am only able to produce



    $sum_{r=1}^{2(i-1)}sum_{k=??}^{??}$ (stuff).



    In particular, I don't see how to reconcile the inequalities $1 leq rleq 2k,$ and $1 leq k leq i-1$, without introducing half integers into the sum.



    Thanks,






    summation







    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      I would like to exchange the order of the following:



      $sum_{k=1}^{i-1}sum_{r=1}^{2k}$ (stuff).



      I feel like it should be easy, but so far I am only able to produce



      $sum_{r=1}^{2(i-1)}sum_{k=??}^{??}$ (stuff).



      In particular, I don't see how to reconcile the inequalities $1 leq rleq 2k,$ and $1 leq k leq i-1$, without introducing half integers into the sum.



      Thanks,






      summation







      share|cite|improve this question









      $endgroup$




      I would like to exchange the order of the following:



      $sum_{k=1}^{i-1}sum_{r=1}^{2k}$ (stuff).



      I feel like it should be easy, but so far I am only able to produce



      $sum_{r=1}^{2(i-1)}sum_{k=??}^{??}$ (stuff).



      In particular, I don't see how to reconcile the inequalities $1 leq rleq 2k,$ and $1 leq k leq i-1$, without introducing half integers into the sum.



      Thanks,






      summation




      summation






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 8 at 6:05









      MathIsArtMathIsArt

      1328




      1328






















          1 Answer
          1






          active

          oldest

          votes


















          2












          $begingroup$

          The second sum should be $$sum_{k=lceil frac r2rceil}^{i-1}$$
          $k$ is always at least $frac r2$ and ranges up to $i-1$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I think $k$ starts from $[frac r 2]$ if $r$ is even and $[frac r 2]+1$ if $r$ is odd.
            $endgroup$
            – Kavi Rama Murthy
            Jan 8 at 6:17






          • 1




            $begingroup$
            @Kavi That's exactly what $lceil{rover2}rceil$ means.
            $endgroup$
            – Ivan Neretin
            Jan 8 at 6:23










          • $begingroup$
            I see. I didn't know this. Thanks.
            $endgroup$
            – Kavi Rama Murthy
            Jan 8 at 6:29










          • $begingroup$
            Your answer is clear, thanks.
            $endgroup$
            – MathIsArt
            Jan 8 at 7:06










          • $begingroup$
            @KaviRamaMurthy: those brackets represent the ceiling function-you round up to the next integer. Upside down ones are the floor, where you round down.
            $endgroup$
            – Ross Millikan
            Jan 8 at 15:02












          Your Answer





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          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          2












          $begingroup$

          The second sum should be $$sum_{k=lceil frac r2rceil}^{i-1}$$
          $k$ is always at least $frac r2$ and ranges up to $i-1$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I think $k$ starts from $[frac r 2]$ if $r$ is even and $[frac r 2]+1$ if $r$ is odd.
            $endgroup$
            – Kavi Rama Murthy
            Jan 8 at 6:17






          • 1




            $begingroup$
            @Kavi That's exactly what $lceil{rover2}rceil$ means.
            $endgroup$
            – Ivan Neretin
            Jan 8 at 6:23










          • $begingroup$
            I see. I didn't know this. Thanks.
            $endgroup$
            – Kavi Rama Murthy
            Jan 8 at 6:29










          • $begingroup$
            Your answer is clear, thanks.
            $endgroup$
            – MathIsArt
            Jan 8 at 7:06










          • $begingroup$
            @KaviRamaMurthy: those brackets represent the ceiling function-you round up to the next integer. Upside down ones are the floor, where you round down.
            $endgroup$
            – Ross Millikan
            Jan 8 at 15:02
















          2












          $begingroup$

          The second sum should be $$sum_{k=lceil frac r2rceil}^{i-1}$$
          $k$ is always at least $frac r2$ and ranges up to $i-1$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I think $k$ starts from $[frac r 2]$ if $r$ is even and $[frac r 2]+1$ if $r$ is odd.
            $endgroup$
            – Kavi Rama Murthy
            Jan 8 at 6:17






          • 1




            $begingroup$
            @Kavi That's exactly what $lceil{rover2}rceil$ means.
            $endgroup$
            – Ivan Neretin
            Jan 8 at 6:23










          • $begingroup$
            I see. I didn't know this. Thanks.
            $endgroup$
            – Kavi Rama Murthy
            Jan 8 at 6:29










          • $begingroup$
            Your answer is clear, thanks.
            $endgroup$
            – MathIsArt
            Jan 8 at 7:06










          • $begingroup$
            @KaviRamaMurthy: those brackets represent the ceiling function-you round up to the next integer. Upside down ones are the floor, where you round down.
            $endgroup$
            – Ross Millikan
            Jan 8 at 15:02














          2












          2








          2





          $begingroup$

          The second sum should be $$sum_{k=lceil frac r2rceil}^{i-1}$$
          $k$ is always at least $frac r2$ and ranges up to $i-1$






          share|cite|improve this answer









          $endgroup$



          The second sum should be $$sum_{k=lceil frac r2rceil}^{i-1}$$
          $k$ is always at least $frac r2$ and ranges up to $i-1$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 8 at 6:12









          Ross MillikanRoss Millikan

          300k24200375




          300k24200375












          • $begingroup$
            I think $k$ starts from $[frac r 2]$ if $r$ is even and $[frac r 2]+1$ if $r$ is odd.
            $endgroup$
            – Kavi Rama Murthy
            Jan 8 at 6:17






          • 1




            $begingroup$
            @Kavi That's exactly what $lceil{rover2}rceil$ means.
            $endgroup$
            – Ivan Neretin
            Jan 8 at 6:23










          • $begingroup$
            I see. I didn't know this. Thanks.
            $endgroup$
            – Kavi Rama Murthy
            Jan 8 at 6:29










          • $begingroup$
            Your answer is clear, thanks.
            $endgroup$
            – MathIsArt
            Jan 8 at 7:06










          • $begingroup$
            @KaviRamaMurthy: those brackets represent the ceiling function-you round up to the next integer. Upside down ones are the floor, where you round down.
            $endgroup$
            – Ross Millikan
            Jan 8 at 15:02


















          • $begingroup$
            I think $k$ starts from $[frac r 2]$ if $r$ is even and $[frac r 2]+1$ if $r$ is odd.
            $endgroup$
            – Kavi Rama Murthy
            Jan 8 at 6:17






          • 1




            $begingroup$
            @Kavi That's exactly what $lceil{rover2}rceil$ means.
            $endgroup$
            – Ivan Neretin
            Jan 8 at 6:23










          • $begingroup$
            I see. I didn't know this. Thanks.
            $endgroup$
            – Kavi Rama Murthy
            Jan 8 at 6:29










          • $begingroup$
            Your answer is clear, thanks.
            $endgroup$
            – MathIsArt
            Jan 8 at 7:06










          • $begingroup$
            @KaviRamaMurthy: those brackets represent the ceiling function-you round up to the next integer. Upside down ones are the floor, where you round down.
            $endgroup$
            – Ross Millikan
            Jan 8 at 15:02
















          $begingroup$
          I think $k$ starts from $[frac r 2]$ if $r$ is even and $[frac r 2]+1$ if $r$ is odd.
          $endgroup$
          – Kavi Rama Murthy
          Jan 8 at 6:17




          $begingroup$
          I think $k$ starts from $[frac r 2]$ if $r$ is even and $[frac r 2]+1$ if $r$ is odd.
          $endgroup$
          – Kavi Rama Murthy
          Jan 8 at 6:17




          1




          1




          $begingroup$
          @Kavi That's exactly what $lceil{rover2}rceil$ means.
          $endgroup$
          – Ivan Neretin
          Jan 8 at 6:23




          $begingroup$
          @Kavi That's exactly what $lceil{rover2}rceil$ means.
          $endgroup$
          – Ivan Neretin
          Jan 8 at 6:23












          $begingroup$
          I see. I didn't know this. Thanks.
          $endgroup$
          – Kavi Rama Murthy
          Jan 8 at 6:29




          $begingroup$
          I see. I didn't know this. Thanks.
          $endgroup$
          – Kavi Rama Murthy
          Jan 8 at 6:29












          $begingroup$
          Your answer is clear, thanks.
          $endgroup$
          – MathIsArt
          Jan 8 at 7:06




          $begingroup$
          Your answer is clear, thanks.
          $endgroup$
          – MathIsArt
          Jan 8 at 7:06












          $begingroup$
          @KaviRamaMurthy: those brackets represent the ceiling function-you round up to the next integer. Upside down ones are the floor, where you round down.
          $endgroup$
          – Ross Millikan
          Jan 8 at 15:02




          $begingroup$
          @KaviRamaMurthy: those brackets represent the ceiling function-you round up to the next integer. Upside down ones are the floor, where you round down.
          $endgroup$
          – Ross Millikan
          Jan 8 at 15:02


















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