Evaluate the limit by first recognizing the sum as a Riemann Sum for a function defined on $[0,1]$.












0












$begingroup$


Evaluate the limit by first recognizing the sum as a Riemann Sum for a function defined on $[0,1]$
$$
limlimits_{n to infty} sumlimits_{i=1}^{n} left(frac{i^5}{n^6}+frac{i}{n^2}right).
$$



I understand how to do a Riemann Sum but am not sure how to find f(x) from the limit.



I apologize for any errors in the way I've posted this. This is my first question on here so I don't quite know the swing of things yet.
Thanks everyone for your help!






calculus limits riemann-sum







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    0












    $begingroup$


    Evaluate the limit by first recognizing the sum as a Riemann Sum for a function defined on $[0,1]$
    $$
    limlimits_{n to infty} sumlimits_{i=1}^{n} left(frac{i^5}{n^6}+frac{i}{n^2}right).
    $$



    I understand how to do a Riemann Sum but am not sure how to find f(x) from the limit.



    I apologize for any errors in the way I've posted this. This is my first question on here so I don't quite know the swing of things yet.
    Thanks everyone for your help!






    calculus limits riemann-sum







    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      Evaluate the limit by first recognizing the sum as a Riemann Sum for a function defined on $[0,1]$
      $$
      limlimits_{n to infty} sumlimits_{i=1}^{n} left(frac{i^5}{n^6}+frac{i}{n^2}right).
      $$



      I understand how to do a Riemann Sum but am not sure how to find f(x) from the limit.



      I apologize for any errors in the way I've posted this. This is my first question on here so I don't quite know the swing of things yet.
      Thanks everyone for your help!






      calculus limits riemann-sum







      share|cite|improve this question











      $endgroup$




      Evaluate the limit by first recognizing the sum as a Riemann Sum for a function defined on $[0,1]$
      $$
      limlimits_{n to infty} sumlimits_{i=1}^{n} left(frac{i^5}{n^6}+frac{i}{n^2}right).
      $$



      I understand how to do a Riemann Sum but am not sure how to find f(x) from the limit.



      I apologize for any errors in the way I've posted this. This is my first question on here so I don't quite know the swing of things yet.
      Thanks everyone for your help!






      calculus limits riemann-sum




      calculus limits riemann-sum






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      share|cite|improve this question













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      edited Feb 1 at 21:16







      user229257

















      asked Feb 1 at 19:07









      user229257user229257

      154




      154






















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          $begingroup$

          If you pull out $frac{1}{n}$ you get
          $$
          frac{1}{n}sum_{i = 1}^{n} left[left(frac{i}{n}right)^5 + left(frac{i}{n}right)right],
          $$
          which is a Riemann sum for the function $f(x)=x^5+x$ on the interval $[0,1]$.
          Your limit thus equals
          $$
          intlimits_0^1(x^5+x)dx
          = left[ frac{x^6}{6} + frac{x^2}{2} right]_{x = 0}^{1}
          = frac{1}{6} + frac{1}{2}
          = frac{2}{3}.
          $$






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            $begingroup$

            If you pull out $frac{1}{n}$ you get
            $$
            frac{1}{n}sum_{i = 1}^{n} left[left(frac{i}{n}right)^5 + left(frac{i}{n}right)right],
            $$
            which is a Riemann sum for the function $f(x)=x^5+x$ on the interval $[0,1]$.
            Your limit thus equals
            $$
            intlimits_0^1(x^5+x)dx
            = left[ frac{x^6}{6} + frac{x^2}{2} right]_{x = 0}^{1}
            = frac{1}{6} + frac{1}{2}
            = frac{2}{3}.
            $$






            share|cite|improve this answer











            $endgroup$


















              1












              $begingroup$

              If you pull out $frac{1}{n}$ you get
              $$
              frac{1}{n}sum_{i = 1}^{n} left[left(frac{i}{n}right)^5 + left(frac{i}{n}right)right],
              $$
              which is a Riemann sum for the function $f(x)=x^5+x$ on the interval $[0,1]$.
              Your limit thus equals
              $$
              intlimits_0^1(x^5+x)dx
              = left[ frac{x^6}{6} + frac{x^2}{2} right]_{x = 0}^{1}
              = frac{1}{6} + frac{1}{2}
              = frac{2}{3}.
              $$






              share|cite|improve this answer











              $endgroup$
















                1












                1








                1





                $begingroup$

                If you pull out $frac{1}{n}$ you get
                $$
                frac{1}{n}sum_{i = 1}^{n} left[left(frac{i}{n}right)^5 + left(frac{i}{n}right)right],
                $$
                which is a Riemann sum for the function $f(x)=x^5+x$ on the interval $[0,1]$.
                Your limit thus equals
                $$
                intlimits_0^1(x^5+x)dx
                = left[ frac{x^6}{6} + frac{x^2}{2} right]_{x = 0}^{1}
                = frac{1}{6} + frac{1}{2}
                = frac{2}{3}.
                $$






                share|cite|improve this answer











                $endgroup$



                If you pull out $frac{1}{n}$ you get
                $$
                frac{1}{n}sum_{i = 1}^{n} left[left(frac{i}{n}right)^5 + left(frac{i}{n}right)right],
                $$
                which is a Riemann sum for the function $f(x)=x^5+x$ on the interval $[0,1]$.
                Your limit thus equals
                $$
                intlimits_0^1(x^5+x)dx
                = left[ frac{x^6}{6} + frac{x^2}{2} right]_{x = 0}^{1}
                = frac{1}{6} + frac{1}{2}
                = frac{2}{3}.
                $$







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited Feb 1 at 21:02









                Viktor Glombik

                1,3522628




                1,3522628










                answered Feb 1 at 20:32









                GReyesGReyes

                2,43815




                2,43815






























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