Tight upper bound for the difference between two fractions in summations












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I would like to compute the tight upper bound of an error term, $epsilon$, defined as



$epsilon = left |frac{1}{k}sumlimits_{i=1}^{k} frac{x_i}{y_i} - frac{sumlimits_{i=1}^{k} x_i}{sumlimits_{i=1}^{k} y_i} right |$ ,



where $k$, $x_i$ and $y_i$ are non-negative integers such that $x_i leq y_i$. It is easy to see that $sumlimits_{i=1}^{k} frac{x_i}{y_i} leq sumlimits_{i=1}^{k} x_i$. But that does not get me anywhere.






combinatorics inequality summation upper-lower-bounds







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    1












    $begingroup$


    I would like to compute the tight upper bound of an error term, $epsilon$, defined as



    $epsilon = left |frac{1}{k}sumlimits_{i=1}^{k} frac{x_i}{y_i} - frac{sumlimits_{i=1}^{k} x_i}{sumlimits_{i=1}^{k} y_i} right |$ ,



    where $k$, $x_i$ and $y_i$ are non-negative integers such that $x_i leq y_i$. It is easy to see that $sumlimits_{i=1}^{k} frac{x_i}{y_i} leq sumlimits_{i=1}^{k} x_i$. But that does not get me anywhere.






    combinatorics inequality summation upper-lower-bounds







    share|cite|improve this question











    $endgroup$















      1












      1








      1


      0



      $begingroup$


      I would like to compute the tight upper bound of an error term, $epsilon$, defined as



      $epsilon = left |frac{1}{k}sumlimits_{i=1}^{k} frac{x_i}{y_i} - frac{sumlimits_{i=1}^{k} x_i}{sumlimits_{i=1}^{k} y_i} right |$ ,



      where $k$, $x_i$ and $y_i$ are non-negative integers such that $x_i leq y_i$. It is easy to see that $sumlimits_{i=1}^{k} frac{x_i}{y_i} leq sumlimits_{i=1}^{k} x_i$. But that does not get me anywhere.






      combinatorics inequality summation upper-lower-bounds







      share|cite|improve this question











      $endgroup$




      I would like to compute the tight upper bound of an error term, $epsilon$, defined as



      $epsilon = left |frac{1}{k}sumlimits_{i=1}^{k} frac{x_i}{y_i} - frac{sumlimits_{i=1}^{k} x_i}{sumlimits_{i=1}^{k} y_i} right |$ ,



      where $k$, $x_i$ and $y_i$ are non-negative integers such that $x_i leq y_i$. It is easy to see that $sumlimits_{i=1}^{k} frac{x_i}{y_i} leq sumlimits_{i=1}^{k} x_i$. But that does not get me anywhere.






      combinatorics inequality summation upper-lower-bounds




      combinatorics inequality summation upper-lower-bounds






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Feb 1 at 12:25







      consumer

















      asked Jan 31 at 12:57









      consumerconsumer

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          $begingroup$

          Suppose $N$ is a positive integer, $k = N + 1$, $x_1 = 1$, $y_1 = N^2$, and $x_i = y_i = 1$ for all $i > 1$. Then
          $$
          epsilon = frac1{N+1}left(N + frac{1}{N^2}right) - frac{N+1}{N^2 + N} > 1 - frac1{N} - frac1{N+1},
          $$
          so the least upper bound (assuming $k$ is variable) is $1$.






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            $begingroup$

            Suppose $N$ is a positive integer, $k = N + 1$, $x_1 = 1$, $y_1 = N^2$, and $x_i = y_i = 1$ for all $i > 1$. Then
            $$
            epsilon = frac1{N+1}left(N + frac{1}{N^2}right) - frac{N+1}{N^2 + N} > 1 - frac1{N} - frac1{N+1},
            $$
            so the least upper bound (assuming $k$ is variable) is $1$.






            share|cite|improve this answer









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              $begingroup$

              Suppose $N$ is a positive integer, $k = N + 1$, $x_1 = 1$, $y_1 = N^2$, and $x_i = y_i = 1$ for all $i > 1$. Then
              $$
              epsilon = frac1{N+1}left(N + frac{1}{N^2}right) - frac{N+1}{N^2 + N} > 1 - frac1{N} - frac1{N+1},
              $$
              so the least upper bound (assuming $k$ is variable) is $1$.






              share|cite|improve this answer









              $endgroup$
















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                $begingroup$

                Suppose $N$ is a positive integer, $k = N + 1$, $x_1 = 1$, $y_1 = N^2$, and $x_i = y_i = 1$ for all $i > 1$. Then
                $$
                epsilon = frac1{N+1}left(N + frac{1}{N^2}right) - frac{N+1}{N^2 + N} > 1 - frac1{N} - frac1{N+1},
                $$
                so the least upper bound (assuming $k$ is variable) is $1$.






                share|cite|improve this answer









                $endgroup$



                Suppose $N$ is a positive integer, $k = N + 1$, $x_1 = 1$, $y_1 = N^2$, and $x_i = y_i = 1$ for all $i > 1$. Then
                $$
                epsilon = frac1{N+1}left(N + frac{1}{N^2}right) - frac{N+1}{N^2 + N} > 1 - frac1{N} - frac1{N+1},
                $$
                so the least upper bound (assuming $k$ is variable) is $1$.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Feb 1 at 16:20









                Calum GilhooleyCalum Gilhooley

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                5,119730






























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