Evaluating a limit involving an infinite summation

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Evaluate



$$ lim_h to 0 hsum_n=0^infty e^-n^2h^2$$



I think it is somehow related to Riemann Sums, but I'm not sure.

Please help.






calculus limits riemann-sum







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    up vote
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    down vote

    favorite
    2












    Evaluate



    $$ lim_h to 0 hsum_n=0^infty e^-n^2h^2$$



    I think it is somehow related to Riemann Sums, but I'm not sure.

    Please help.






    calculus limits riemann-sum







    share|cite|improve this question























      up vote
      2
      down vote

      favorite
      2









      up vote
      2
      down vote

      favorite
      2






      2





      Evaluate



      $$ lim_h to 0 hsum_n=0^infty e^-n^2h^2$$



      I think it is somehow related to Riemann Sums, but I'm not sure.

      Please help.






      calculus limits riemann-sum







      share|cite|improve this question













      Evaluate



      $$ lim_h to 0 hsum_n=0^infty e^-n^2h^2$$



      I think it is somehow related to Riemann Sums, but I'm not sure.

      Please help.






      calculus limits riemann-sum




      calculus limits riemann-sum






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      share|cite|improve this question











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      asked 43 mins ago









      Robin

      1274




      1274




















          2 Answers
          2






          active

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          up vote
          4
          down vote













          If a function $f$ is Riemann-integrable on $[0,infty)$, then its integral $int_0^infty f(x),dx$ is equal to
          $$lim_hrightarrow 0,hsum_n=0^infty f(nh)$$
          Now you just have to set $f$ appropriately. See this link if you can't do the resulting integration.






          share|cite|improve this answer




















          • What theorem did you use? Can you give a short proof or a reference?
            – Robin
            27 mins ago










          • I used the definition of the Riemann integral.
            – TonyK
            27 mins ago










          • I only know these definitions and properties listed here.
            – Robin
            24 mins ago

















          up vote
          3
          down vote













          I think a good way is to use integral test for convergence.



          This shows that for all $h > 0$, you have :
          $$|sum_n=0^infty e^-n^2 h^2 - int_0^infty e^-x^2 h^2 dx| leq 1$$.
          Using this you can see that
          $$lim_h to 0 h sum_n=0^infty e^-n^2 h^2 = lim_h to 0 h int_0^infty e^-x^2 h^2 dx$$.
          This last integral is a Gaussian integral, and can be evaluated for any $h > 0$:
          $$ int_0^infty e^-x^2 h^2 dx = fracsqrtpi2 frac1h$$



          So finally $$lim_h to 0 h sum_n=0^infty e^-n^2 h^2 = fracsqrtpi2$$






          share|cite|improve this answer








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          seamp is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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            2 Answers
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            active

            oldest

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            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            4
            down vote













            If a function $f$ is Riemann-integrable on $[0,infty)$, then its integral $int_0^infty f(x),dx$ is equal to
            $$lim_hrightarrow 0,hsum_n=0^infty f(nh)$$
            Now you just have to set $f$ appropriately. See this link if you can't do the resulting integration.






            share|cite|improve this answer




















            • What theorem did you use? Can you give a short proof or a reference?
              – Robin
              27 mins ago










            • I used the definition of the Riemann integral.
              – TonyK
              27 mins ago










            • I only know these definitions and properties listed here.
              – Robin
              24 mins ago














            up vote
            4
            down vote













            If a function $f$ is Riemann-integrable on $[0,infty)$, then its integral $int_0^infty f(x),dx$ is equal to
            $$lim_hrightarrow 0,hsum_n=0^infty f(nh)$$
            Now you just have to set $f$ appropriately. See this link if you can't do the resulting integration.






            share|cite|improve this answer




















            • What theorem did you use? Can you give a short proof or a reference?
              – Robin
              27 mins ago










            • I used the definition of the Riemann integral.
              – TonyK
              27 mins ago










            • I only know these definitions and properties listed here.
              – Robin
              24 mins ago












            up vote
            4
            down vote










            up vote
            4
            down vote









            If a function $f$ is Riemann-integrable on $[0,infty)$, then its integral $int_0^infty f(x),dx$ is equal to
            $$lim_hrightarrow 0,hsum_n=0^infty f(nh)$$
            Now you just have to set $f$ appropriately. See this link if you can't do the resulting integration.






            share|cite|improve this answer












            If a function $f$ is Riemann-integrable on $[0,infty)$, then its integral $int_0^infty f(x),dx$ is equal to
            $$lim_hrightarrow 0,hsum_n=0^infty f(nh)$$
            Now you just have to set $f$ appropriately. See this link if you can't do the resulting integration.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered 29 mins ago









            TonyK

            39.7k349128




            39.7k349128











            • What theorem did you use? Can you give a short proof or a reference?
              – Robin
              27 mins ago










            • I used the definition of the Riemann integral.
              – TonyK
              27 mins ago










            • I only know these definitions and properties listed here.
              – Robin
              24 mins ago
















            • What theorem did you use? Can you give a short proof or a reference?
              – Robin
              27 mins ago










            • I used the definition of the Riemann integral.
              – TonyK
              27 mins ago










            • I only know these definitions and properties listed here.
              – Robin
              24 mins ago















            What theorem did you use? Can you give a short proof or a reference?
            – Robin
            27 mins ago




            What theorem did you use? Can you give a short proof or a reference?
            – Robin
            27 mins ago












            I used the definition of the Riemann integral.
            – TonyK
            27 mins ago




            I used the definition of the Riemann integral.
            – TonyK
            27 mins ago












            I only know these definitions and properties listed here.
            – Robin
            24 mins ago




            I only know these definitions and properties listed here.
            – Robin
            24 mins ago










            up vote
            3
            down vote













            I think a good way is to use integral test for convergence.



            This shows that for all $h > 0$, you have :
            $$|sum_n=0^infty e^-n^2 h^2 - int_0^infty e^-x^2 h^2 dx| leq 1$$.
            Using this you can see that
            $$lim_h to 0 h sum_n=0^infty e^-n^2 h^2 = lim_h to 0 h int_0^infty e^-x^2 h^2 dx$$.
            This last integral is a Gaussian integral, and can be evaluated for any $h > 0$:
            $$ int_0^infty e^-x^2 h^2 dx = fracsqrtpi2 frac1h$$



            So finally $$lim_h to 0 h sum_n=0^infty e^-n^2 h^2 = fracsqrtpi2$$






            share|cite|improve this answer








            New contributor




            seamp is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
            Check out our Code of Conduct.





















              up vote
              3
              down vote













              I think a good way is to use integral test for convergence.



              This shows that for all $h > 0$, you have :
              $$|sum_n=0^infty e^-n^2 h^2 - int_0^infty e^-x^2 h^2 dx| leq 1$$.
              Using this you can see that
              $$lim_h to 0 h sum_n=0^infty e^-n^2 h^2 = lim_h to 0 h int_0^infty e^-x^2 h^2 dx$$.
              This last integral is a Gaussian integral, and can be evaluated for any $h > 0$:
              $$ int_0^infty e^-x^2 h^2 dx = fracsqrtpi2 frac1h$$



              So finally $$lim_h to 0 h sum_n=0^infty e^-n^2 h^2 = fracsqrtpi2$$






              share|cite|improve this answer








              New contributor




              seamp is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
              Check out our Code of Conduct.



















                up vote
                3
                down vote










                up vote
                3
                down vote









                I think a good way is to use integral test for convergence.



                This shows that for all $h > 0$, you have :
                $$|sum_n=0^infty e^-n^2 h^2 - int_0^infty e^-x^2 h^2 dx| leq 1$$.
                Using this you can see that
                $$lim_h to 0 h sum_n=0^infty e^-n^2 h^2 = lim_h to 0 h int_0^infty e^-x^2 h^2 dx$$.
                This last integral is a Gaussian integral, and can be evaluated for any $h > 0$:
                $$ int_0^infty e^-x^2 h^2 dx = fracsqrtpi2 frac1h$$



                So finally $$lim_h to 0 h sum_n=0^infty e^-n^2 h^2 = fracsqrtpi2$$






                share|cite|improve this answer








                New contributor




                seamp is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                Check out our Code of Conduct.









                I think a good way is to use integral test for convergence.



                This shows that for all $h > 0$, you have :
                $$|sum_n=0^infty e^-n^2 h^2 - int_0^infty e^-x^2 h^2 dx| leq 1$$.
                Using this you can see that
                $$lim_h to 0 h sum_n=0^infty e^-n^2 h^2 = lim_h to 0 h int_0^infty e^-x^2 h^2 dx$$.
                This last integral is a Gaussian integral, and can be evaluated for any $h > 0$:
                $$ int_0^infty e^-x^2 h^2 dx = fracsqrtpi2 frac1h$$



                So finally $$lim_h to 0 h sum_n=0^infty e^-n^2 h^2 = fracsqrtpi2$$







                share|cite|improve this answer








                New contributor




                seamp is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                Check out our Code of Conduct.









                share|cite|improve this answer



                share|cite|improve this answer






                New contributor




                seamp is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                Check out our Code of Conduct.









                answered 25 mins ago









                seamp

                1314




                1314




                New contributor




                seamp is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                Check out our Code of Conduct.





                New contributor





                seamp is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                Check out our Code of Conduct.






                seamp is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                Check out our Code of Conduct.



























                     

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