How to evaluate $lim_{n to infty} sum_{k=1}^n frac{n+k}{n^3+k}$?











up vote
0
down vote

favorite













How to evaluate the following limit?
$$lim_{n to infty} sum_{k=1}^n frac{n+k}{n^3+k}$$




I think that every term of the sum is greater than the first one and smaller than the last one and then from the squeeze theorem the limit is $0$. However, I can't prove that inequality.






calculus sequences-and-series limits







share|cite|improve this question




















  • 1




    What are you asking?
    – Will M.
    Nov 19 at 18:31






  • 1




    The sum is less than $n frac{n +n}{n^3+1}$
    – RRL
    Nov 19 at 18:33

















up vote
0
down vote

favorite













How to evaluate the following limit?
$$lim_{n to infty} sum_{k=1}^n frac{n+k}{n^3+k}$$




I think that every term of the sum is greater than the first one and smaller than the last one and then from the squeeze theorem the limit is $0$. However, I can't prove that inequality.






calculus sequences-and-series limits







share|cite|improve this question




















  • 1




    What are you asking?
    – Will M.
    Nov 19 at 18:31






  • 1




    The sum is less than $n frac{n +n}{n^3+1}$
    – RRL
    Nov 19 at 18:33















up vote
0
down vote

favorite









up vote
0
down vote

favorite












How to evaluate the following limit?
$$lim_{n to infty} sum_{k=1}^n frac{n+k}{n^3+k}$$




I think that every term of the sum is greater than the first one and smaller than the last one and then from the squeeze theorem the limit is $0$. However, I can't prove that inequality.






calculus sequences-and-series limits







share|cite|improve this question
















How to evaluate the following limit?
$$lim_{n to infty} sum_{k=1}^n frac{n+k}{n^3+k}$$




I think that every term of the sum is greater than the first one and smaller than the last one and then from the squeeze theorem the limit is $0$. However, I can't prove that inequality.






calculus sequences-and-series limits




calculus sequences-and-series limits






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 2 days ago









user587192

1,21810




1,21810










asked Nov 19 at 18:27









user69503

486




486








  • 1




    What are you asking?
    – Will M.
    Nov 19 at 18:31






  • 1




    The sum is less than $n frac{n +n}{n^3+1}$
    – RRL
    Nov 19 at 18:33
















  • 1




    What are you asking?
    – Will M.
    Nov 19 at 18:31






  • 1




    The sum is less than $n frac{n +n}{n^3+1}$
    – RRL
    Nov 19 at 18:33










1




1




What are you asking?
– Will M.
Nov 19 at 18:31




What are you asking?
– Will M.
Nov 19 at 18:31




1




1




The sum is less than $n frac{n +n}{n^3+1}$
– RRL
Nov 19 at 18:33






The sum is less than $n frac{n +n}{n^3+1}$
– RRL
Nov 19 at 18:33












2 Answers
2






active

oldest

votes

















up vote
5
down vote



accepted










This should work:
$$sum_{k=1}^nfrac{n+k}{n^3+k}leqsum_{k=1}^nfrac{n+k}{n^3}leqsum_{k=1}^nfrac{n+n}{n^3}=frac{2n^2}{n^3}=frac{2}{n}to0$$



Also, every term is $geq 0$ so $0$ is also a lower bound.






share|cite|improve this answer




























    up vote
    1
    down vote













    HINT:



    Note that $n+1le n+kle 2n$ and $n^3+1le n^3 +kle n^3+n$.






    share|cite|improve this answer





















      Your Answer





      StackExchange.ifUsing("editor", function () {
      return StackExchange.using("mathjaxEditing", function () {
      StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
      StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
      });
      });
      }, "mathjax-editing");

      StackExchange.ready(function() {
      var channelOptions = {
      tags: "".split(" "),
      id: "69"
      };
      initTagRenderer("".split(" "), "".split(" "), channelOptions);

      StackExchange.using("externalEditor", function() {
      // Have to fire editor after snippets, if snippets enabled
      if (StackExchange.settings.snippets.snippetsEnabled) {
      StackExchange.using("snippets", function() {
      createEditor();
      });
      }
      else {
      createEditor();
      }
      });

      function createEditor() {
      StackExchange.prepareEditor({
      heartbeatType: 'answer',
      convertImagesToLinks: true,
      noModals: true,
      showLowRepImageUploadWarning: true,
      reputationToPostImages: 10,
      bindNavPrevention: true,
      postfix: "",
      imageUploader: {
      brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
      contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
      allowUrls: true
      },
      noCode: true, onDemand: true,
      discardSelector: ".discard-answer"
      ,immediatelyShowMarkdownHelp:true
      });


      }
      });














       

      draft saved


      draft discarded


















      StackExchange.ready(
      function () {
      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3005314%2fhow-to-evaluate-lim-n-to-infty-sum-k-1n-fracnkn3k%23new-answer', 'question_page');
      }
      );

      Post as a guest















      Required, but never shown

























      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes








      up vote
      5
      down vote



      accepted










      This should work:
      $$sum_{k=1}^nfrac{n+k}{n^3+k}leqsum_{k=1}^nfrac{n+k}{n^3}leqsum_{k=1}^nfrac{n+n}{n^3}=frac{2n^2}{n^3}=frac{2}{n}to0$$



      Also, every term is $geq 0$ so $0$ is also a lower bound.






      share|cite|improve this answer

























        up vote
        5
        down vote



        accepted










        This should work:
        $$sum_{k=1}^nfrac{n+k}{n^3+k}leqsum_{k=1}^nfrac{n+k}{n^3}leqsum_{k=1}^nfrac{n+n}{n^3}=frac{2n^2}{n^3}=frac{2}{n}to0$$



        Also, every term is $geq 0$ so $0$ is also a lower bound.






        share|cite|improve this answer























          up vote
          5
          down vote



          accepted







          up vote
          5
          down vote



          accepted






          This should work:
          $$sum_{k=1}^nfrac{n+k}{n^3+k}leqsum_{k=1}^nfrac{n+k}{n^3}leqsum_{k=1}^nfrac{n+n}{n^3}=frac{2n^2}{n^3}=frac{2}{n}to0$$



          Also, every term is $geq 0$ so $0$ is also a lower bound.






          share|cite|improve this answer












          This should work:
          $$sum_{k=1}^nfrac{n+k}{n^3+k}leqsum_{k=1}^nfrac{n+k}{n^3}leqsum_{k=1}^nfrac{n+n}{n^3}=frac{2n^2}{n^3}=frac{2}{n}to0$$



          Also, every term is $geq 0$ so $0$ is also a lower bound.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 19 at 18:33









          Marco

          1909




          1909






















              up vote
              1
              down vote













              HINT:



              Note that $n+1le n+kle 2n$ and $n^3+1le n^3 +kle n^3+n$.






              share|cite|improve this answer

























                up vote
                1
                down vote













                HINT:



                Note that $n+1le n+kle 2n$ and $n^3+1le n^3 +kle n^3+n$.






                share|cite|improve this answer























                  up vote
                  1
                  down vote










                  up vote
                  1
                  down vote









                  HINT:



                  Note that $n+1le n+kle 2n$ and $n^3+1le n^3 +kle n^3+n$.






                  share|cite|improve this answer












                  HINT:



                  Note that $n+1le n+kle 2n$ and $n^3+1le n^3 +kle n^3+n$.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Nov 19 at 18:35









                  Mark Viola

                  129k1273170




                  129k1273170






























                       

                      draft saved


                      draft discarded



















































                       


                      draft saved


                      draft discarded














                      StackExchange.ready(
                      function () {
                      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3005314%2fhow-to-evaluate-lim-n-to-infty-sum-k-1n-fracnkn3k%23new-answer', 'question_page');
                      }
                      );

                      Post as a guest















                      Required, but never shown





















































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown

































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown







                      Popular posts from this blog

                      'app-layout' is not a known element: how to share Component with different Modules

                      Image
                      0 My application was working fine until I decided to go with lazy loading: So, my shared component looks like this: import { Component, Renderer2 } from '@angular/core'; export interface FormModel { captcha?: string; } @Component({ selector: 'app-layout', templateUrl: './app-layout.component.html', styleUrls: ['./app-layout.css'] }) export class FullLayoutComponent { } app-layout.component.ts I am using this component in my Another component say school-home.component.html like this: <app-layout> </app-layout> and it was working fine until I created a Module school-home.module.ts like this: import { NgModule } from '@angular/core'; import { CommonModule } from '@angular/common'; import { SchoolhomeRoutingModule ...
                      Read more

                      android studio warns about leanback feature tag usage required on manifest while using Unity exported app?

                      Image
                      .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ height:90px;width:728px;box-sizing:border-box; } 0 This is a simple project showing cubes on Unity, exported to android project and imported on Android studio. On maifest file it showed following warning: Expecting <uses-feature android:name="android.software.leanback" android:required="false" /> tag I have no information about it. And also on default emptyview android project does not use/mention this feature. I know it is required by TV's and such but why its not added by default? Why is this? And Should I concern about it? android unity3d ...
                      Read more

                      WPF add header to Image with URL pettitions [duplicate]

                      Image
                      up vote 0 down vote favorite This question already has an answer here: Image source URI requiring authorization 3 answers Add header to all* HTTP requests on a machine 1 answer first of all sorry for my english. Recently I added some security in the backend of my project, and now I need to send a validation code(apiKey) inside the headers to retrieve the images. I have a problem with ALL images in my project that I had attached in my UI. For example, before I had: <Image Height="90" Source="{Binding Video.Thumbnail}"/> Now I need to change my code to create a CUSTOM petition add the apiKey in the header:...
                      Read more