Does this power sequence converge or diverge? If it converges, what is the limit?












2












$begingroup$


Say I have this sequence:



$$a_n = frac{n^2}{sqrt{n^3 + 4n}}$$



Again, I don't think I can divide the numerator and denominator by $n^{1.5}$... that seems like it complicates things. What else can I do?



I can't square the top and bottom because that changes the value of the general sequence. Can I divide by $n^2$?



Is this valid:



$$a_n = frac{1}{sqrt{frac{n^3}{n^4} + frac{4}{n}}}$$






sequences-and-series







share|cite|improve this question











$endgroup$












  • $begingroup$
    What are you trying to do with the sequence? Are you trying to determine if it converges / find its limit? In your last identity, you should have $4/n^3$ in the denominator.
    $endgroup$
    – MisterRiemann
    Mar 27 at 16:42


















2












$begingroup$


Say I have this sequence:



$$a_n = frac{n^2}{sqrt{n^3 + 4n}}$$



Again, I don't think I can divide the numerator and denominator by $n^{1.5}$... that seems like it complicates things. What else can I do?



I can't square the top and bottom because that changes the value of the general sequence. Can I divide by $n^2$?



Is this valid:



$$a_n = frac{1}{sqrt{frac{n^3}{n^4} + frac{4}{n}}}$$






sequences-and-series







share|cite|improve this question











$endgroup$












  • $begingroup$
    What are you trying to do with the sequence? Are you trying to determine if it converges / find its limit? In your last identity, you should have $4/n^3$ in the denominator.
    $endgroup$
    – MisterRiemann
    Mar 27 at 16:42
















2












2








2





$begingroup$


Say I have this sequence:



$$a_n = frac{n^2}{sqrt{n^3 + 4n}}$$



Again, I don't think I can divide the numerator and denominator by $n^{1.5}$... that seems like it complicates things. What else can I do?



I can't square the top and bottom because that changes the value of the general sequence. Can I divide by $n^2$?



Is this valid:



$$a_n = frac{1}{sqrt{frac{n^3}{n^4} + frac{4}{n}}}$$






sequences-and-series







share|cite|improve this question











$endgroup$




Say I have this sequence:



$$a_n = frac{n^2}{sqrt{n^3 + 4n}}$$



Again, I don't think I can divide the numerator and denominator by $n^{1.5}$... that seems like it complicates things. What else can I do?



I can't square the top and bottom because that changes the value of the general sequence. Can I divide by $n^2$?



Is this valid:



$$a_n = frac{1}{sqrt{frac{n^3}{n^4} + frac{4}{n}}}$$






sequences-and-series




sequences-and-series






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Mar 27 at 16:44







Jwan622

















asked Mar 27 at 16:41









Jwan622Jwan622

2,34611632




2,34611632












  • $begingroup$
    What are you trying to do with the sequence? Are you trying to determine if it converges / find its limit? In your last identity, you should have $4/n^3$ in the denominator.
    $endgroup$
    – MisterRiemann
    Mar 27 at 16:42




















  • $begingroup$
    What are you trying to do with the sequence? Are you trying to determine if it converges / find its limit? In your last identity, you should have $4/n^3$ in the denominator.
    $endgroup$
    – MisterRiemann
    Mar 27 at 16:42


















$begingroup$
What are you trying to do with the sequence? Are you trying to determine if it converges / find its limit? In your last identity, you should have $4/n^3$ in the denominator.
$endgroup$
– MisterRiemann
Mar 27 at 16:42






$begingroup$
What are you trying to do with the sequence? Are you trying to determine if it converges / find its limit? In your last identity, you should have $4/n^3$ in the denominator.
$endgroup$
– MisterRiemann
Mar 27 at 16:42












3 Answers
3






active

oldest

votes


















3












$begingroup$

You can easily find a divergent minorant:



$$frac{n^2}{sqrt{n^3 + 4n}} ge frac{n^2}{sqrt{n^3 + 4n^color{blue}{3}}} = sqrt{frac{n}{5}} to +infty$$






share|cite|improve this answer









$endgroup$





















    3












    $begingroup$

    Hint: It is $$sqrt{frac{n^4}{n^3+4n}}$$ and this is divergent.






    share|cite|improve this answer











    $endgroup$





















      1












      $begingroup$

      We have:



      $$a_n = frac{sqrt{n} }{sqrt{1 + frac{4}{n^2}}}$$



      You can see that the denominator tends to 1, so that $a_n$ clearly diverges, behaving asymptotically as $sqrt{n}$.






      share|cite|improve this answer









      $endgroup$













      • $begingroup$
        How did you get to here?
        $endgroup$
        – Jwan622
        Mar 27 at 16:49










      • $begingroup$
        Multiply by $frac{n^{1.5}}{n^{1.5}}$.
        $endgroup$
        – MatthewPeter
        Mar 27 at 16:50










      • $begingroup$
        Can you flesh that out a bit? Don't you mean divide top and bottom by $n^{1.5}$
        $endgroup$
        – Jwan622
        Mar 27 at 17:23














      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',
      autoActivateHeartbeat: false,
      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%2f3164743%2fdoes-this-power-sequence-converge-or-diverge-if-it-converges-what-is-the-limit%23new-answer', 'question_page');
      }
      );

      Post as a guest















      Required, but never shown

























      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      3












      $begingroup$

      You can easily find a divergent minorant:



      $$frac{n^2}{sqrt{n^3 + 4n}} ge frac{n^2}{sqrt{n^3 + 4n^color{blue}{3}}} = sqrt{frac{n}{5}} to +infty$$






      share|cite|improve this answer









      $endgroup$


















        3












        $begingroup$

        You can easily find a divergent minorant:



        $$frac{n^2}{sqrt{n^3 + 4n}} ge frac{n^2}{sqrt{n^3 + 4n^color{blue}{3}}} = sqrt{frac{n}{5}} to +infty$$






        share|cite|improve this answer









        $endgroup$
















          3












          3








          3





          $begingroup$

          You can easily find a divergent minorant:



          $$frac{n^2}{sqrt{n^3 + 4n}} ge frac{n^2}{sqrt{n^3 + 4n^color{blue}{3}}} = sqrt{frac{n}{5}} to +infty$$






          share|cite|improve this answer









          $endgroup$



          You can easily find a divergent minorant:



          $$frac{n^2}{sqrt{n^3 + 4n}} ge frac{n^2}{sqrt{n^3 + 4n^color{blue}{3}}} = sqrt{frac{n}{5}} to +infty$$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Mar 27 at 16:46









          StackTDStackTD

          24.5k2254




          24.5k2254























              3












              $begingroup$

              Hint: It is $$sqrt{frac{n^4}{n^3+4n}}$$ and this is divergent.






              share|cite|improve this answer











              $endgroup$


















                3












                $begingroup$

                Hint: It is $$sqrt{frac{n^4}{n^3+4n}}$$ and this is divergent.






                share|cite|improve this answer











                $endgroup$
















                  3












                  3








                  3





                  $begingroup$

                  Hint: It is $$sqrt{frac{n^4}{n^3+4n}}$$ and this is divergent.






                  share|cite|improve this answer











                  $endgroup$



                  Hint: It is $$sqrt{frac{n^4}{n^3+4n}}$$ and this is divergent.







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited Mar 27 at 16:54

























                  answered Mar 27 at 16:46









                  Dr. Sonnhard GraubnerDr. Sonnhard Graubner

                  78.6k42867




                  78.6k42867























                      1












                      $begingroup$

                      We have:



                      $$a_n = frac{sqrt{n} }{sqrt{1 + frac{4}{n^2}}}$$



                      You can see that the denominator tends to 1, so that $a_n$ clearly diverges, behaving asymptotically as $sqrt{n}$.






                      share|cite|improve this answer









                      $endgroup$













                      • $begingroup$
                        How did you get to here?
                        $endgroup$
                        – Jwan622
                        Mar 27 at 16:49










                      • $begingroup$
                        Multiply by $frac{n^{1.5}}{n^{1.5}}$.
                        $endgroup$
                        – MatthewPeter
                        Mar 27 at 16:50










                      • $begingroup$
                        Can you flesh that out a bit? Don't you mean divide top and bottom by $n^{1.5}$
                        $endgroup$
                        – Jwan622
                        Mar 27 at 17:23


















                      1












                      $begingroup$

                      We have:



                      $$a_n = frac{sqrt{n} }{sqrt{1 + frac{4}{n^2}}}$$



                      You can see that the denominator tends to 1, so that $a_n$ clearly diverges, behaving asymptotically as $sqrt{n}$.






                      share|cite|improve this answer









                      $endgroup$













                      • $begingroup$
                        How did you get to here?
                        $endgroup$
                        – Jwan622
                        Mar 27 at 16:49










                      • $begingroup$
                        Multiply by $frac{n^{1.5}}{n^{1.5}}$.
                        $endgroup$
                        – MatthewPeter
                        Mar 27 at 16:50










                      • $begingroup$
                        Can you flesh that out a bit? Don't you mean divide top and bottom by $n^{1.5}$
                        $endgroup$
                        – Jwan622
                        Mar 27 at 17:23
















                      1












                      1








                      1





                      $begingroup$

                      We have:



                      $$a_n = frac{sqrt{n} }{sqrt{1 + frac{4}{n^2}}}$$



                      You can see that the denominator tends to 1, so that $a_n$ clearly diverges, behaving asymptotically as $sqrt{n}$.






                      share|cite|improve this answer









                      $endgroup$



                      We have:



                      $$a_n = frac{sqrt{n} }{sqrt{1 + frac{4}{n^2}}}$$



                      You can see that the denominator tends to 1, so that $a_n$ clearly diverges, behaving asymptotically as $sqrt{n}$.







                      share|cite|improve this answer












                      share|cite|improve this answer



                      share|cite|improve this answer










                      answered Mar 27 at 16:47









                      MatthewPeterMatthewPeter

                      191




                      191












                      • $begingroup$
                        How did you get to here?
                        $endgroup$
                        – Jwan622
                        Mar 27 at 16:49










                      • $begingroup$
                        Multiply by $frac{n^{1.5}}{n^{1.5}}$.
                        $endgroup$
                        – MatthewPeter
                        Mar 27 at 16:50










                      • $begingroup$
                        Can you flesh that out a bit? Don't you mean divide top and bottom by $n^{1.5}$
                        $endgroup$
                        – Jwan622
                        Mar 27 at 17:23




















                      • $begingroup$
                        How did you get to here?
                        $endgroup$
                        – Jwan622
                        Mar 27 at 16:49










                      • $begingroup$
                        Multiply by $frac{n^{1.5}}{n^{1.5}}$.
                        $endgroup$
                        – MatthewPeter
                        Mar 27 at 16:50










                      • $begingroup$
                        Can you flesh that out a bit? Don't you mean divide top and bottom by $n^{1.5}$
                        $endgroup$
                        – Jwan622
                        Mar 27 at 17:23


















                      $begingroup$
                      How did you get to here?
                      $endgroup$
                      – Jwan622
                      Mar 27 at 16:49




                      $begingroup$
                      How did you get to here?
                      $endgroup$
                      – Jwan622
                      Mar 27 at 16:49












                      $begingroup$
                      Multiply by $frac{n^{1.5}}{n^{1.5}}$.
                      $endgroup$
                      – MatthewPeter
                      Mar 27 at 16:50




                      $begingroup$
                      Multiply by $frac{n^{1.5}}{n^{1.5}}$.
                      $endgroup$
                      – MatthewPeter
                      Mar 27 at 16:50












                      $begingroup$
                      Can you flesh that out a bit? Don't you mean divide top and bottom by $n^{1.5}$
                      $endgroup$
                      – Jwan622
                      Mar 27 at 17:23






                      $begingroup$
                      Can you flesh that out a bit? Don't you mean divide top and bottom by $n^{1.5}$
                      $endgroup$
                      – Jwan622
                      Mar 27 at 17:23




















                      draft saved

                      draft discarded




















































                      Thanks for contributing an answer to Mathematics Stack Exchange!


                      • Please be sure to answer the question. Provide details and share your research!

                      But avoid …



                      • Asking for help, clarification, or responding to other answers.

                      • Making statements based on opinion; back them up with references or personal experience.


                      Use MathJax to format equations. MathJax reference.


                      To learn more, see our tips on writing great answers.




                      draft saved


                      draft discarded














                      StackExchange.ready(
                      function () {
                      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3164743%2fdoes-this-power-sequence-converge-or-diverge-if-it-converges-what-is-the-limit%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

                      How to change which sound is reproduced for terminal bell?

                      Image
                      3 Recently upgrade to 18.10 from 18.04, and using the terminal is becoming annoying: the new alert sound is a drip ( /usr/share/sounds/gnome/default/alerts/drip.ogg ), and the previous one is still located at ( /usr/share/sounds/ubuntu/stereo/bell.ogg ). My issue is that at the gnome-terminal, when autocompleting or pressing left BOTH sounds are reproduced, at the same time. I want to just have ubuntu/stereo/bell.ogg active, not both . At the moment I have been only able to reduce the volume of the alerts (but still both are played), or turn the alerts down (but I dislike both "solutions"). gnome-terminal 18.10 teminal-bell share | improve this question
                      Read more

                      Title Spacing in Bjornstrup Chapter, Removing Chapter Number From Contents

                      Image
                      up vote 5 down vote favorite 2 I would like to make it so that the vertical space below the title of each chapter is the same when using the Bjornstrup package (i.e. so that the descenders do not affect the spacing - as shown by the red arrow in the picture below). I think I'm fairly happy with the spacing around "A chapter", so I'd really like titles without descenders (e.g. "Contents") to have the same vertical spacing as "A chapter" below the title. I would also like to remove the large "0" in the contents title. Any help appreciated, as always. documentclass[11pt,a4paper]{book} % Packages[![enter image description here][1]][1] usepackage[left=3.5cm, right=3.5cm, top=4.3cm, bottom=4.25cm]{geometry} usepackage[utf8]{inputenc} usepackage{libertine} usepackage[libertine]{newtx
                      Read more

                      Can I use Tabulator js library in my java Spring + Thymeleaf project?

                      Image
                      0 I want to add Tabulator js to my java project. I added tabulator.min.js, tabulator.min.css to resources folder. After that added to my html page these dependencies as recommending in readme.md. <link rel="stylesheet" th:href="@{/css/tabulator.min.css}"> <script type="text/javascript" th:src="@{/js/tabulator.min.js}"></script> Now my tables view all elements, that I have in the html page, but not correct. In the documentation of Tabulator is example how to create header and table, var table = new Tabulator("#example-table", { columns:[ {title:"Name", field:"name", sorter:"string", width:200, editor:true}, {title:"Age", field:"age", sorter:"
                      Read more