Measure theory integral












4














This is a qualifying exam practice question - so not being graded for homework purposes, just studying!



Calculate $lim_{n rightarrow infty} int_0^infty frac{x^n}{ x^{(n+3)}+1} dx$



I tried the following:



$lim_{n rightarrow infty} int_0^infty frac{x^n}{ x^{(n+3)}+1} , dx$ = $frac{d}{dn}int_0^infty int_0^inftyfrac{x^n}{ x^{(n+3)}+1}dn , dx$ = -$frac{d}{dn} int_0^infty frac{ln(x^3+1}{x^3 ln(x)} , dx$



Not really sure where to go from here, any advice would be appreciated!






real-analysis measure-theory convergence lebesgue-integral lebesgue-measure







share|cite|improve this question




















  • 3




    I'm trying to understand your thinking when you went from $lim_{ntoinfty}$ to $frac{d}{dn}int_0^infty dn$. I don't see how that could work.
    – Teepeemm
    yesterday










  • (Another take on Teepeemm's comment:) $frac{mathrm{d}}{mathrm{d}n} langle text{expression containing no "$n$"s} rangle = 0$.
    – Eric Towers
    yesterday
















4














This is a qualifying exam practice question - so not being graded for homework purposes, just studying!



Calculate $lim_{n rightarrow infty} int_0^infty frac{x^n}{ x^{(n+3)}+1} dx$



I tried the following:



$lim_{n rightarrow infty} int_0^infty frac{x^n}{ x^{(n+3)}+1} , dx$ = $frac{d}{dn}int_0^infty int_0^inftyfrac{x^n}{ x^{(n+3)}+1}dn , dx$ = -$frac{d}{dn} int_0^infty frac{ln(x^3+1}{x^3 ln(x)} , dx$



Not really sure where to go from here, any advice would be appreciated!






real-analysis measure-theory convergence lebesgue-integral lebesgue-measure







share|cite|improve this question




















  • 3




    I'm trying to understand your thinking when you went from $lim_{ntoinfty}$ to $frac{d}{dn}int_0^infty dn$. I don't see how that could work.
    – Teepeemm
    yesterday










  • (Another take on Teepeemm's comment:) $frac{mathrm{d}}{mathrm{d}n} langle text{expression containing no "$n$"s} rangle = 0$.
    – Eric Towers
    yesterday














4












4








4


3





This is a qualifying exam practice question - so not being graded for homework purposes, just studying!



Calculate $lim_{n rightarrow infty} int_0^infty frac{x^n}{ x^{(n+3)}+1} dx$



I tried the following:



$lim_{n rightarrow infty} int_0^infty frac{x^n}{ x^{(n+3)}+1} , dx$ = $frac{d}{dn}int_0^infty int_0^inftyfrac{x^n}{ x^{(n+3)}+1}dn , dx$ = -$frac{d}{dn} int_0^infty frac{ln(x^3+1}{x^3 ln(x)} , dx$



Not really sure where to go from here, any advice would be appreciated!






real-analysis measure-theory convergence lebesgue-integral lebesgue-measure







share|cite|improve this question















This is a qualifying exam practice question - so not being graded for homework purposes, just studying!



Calculate $lim_{n rightarrow infty} int_0^infty frac{x^n}{ x^{(n+3)}+1} dx$



I tried the following:



$lim_{n rightarrow infty} int_0^infty frac{x^n}{ x^{(n+3)}+1} , dx$ = $frac{d}{dn}int_0^infty int_0^inftyfrac{x^n}{ x^{(n+3)}+1}dn , dx$ = -$frac{d}{dn} int_0^infty frac{ln(x^3+1}{x^3 ln(x)} , dx$



Not really sure where to go from here, any advice would be appreciated!






real-analysis measure-theory convergence lebesgue-integral lebesgue-measure




real-analysis measure-theory convergence lebesgue-integral lebesgue-measure






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited yesterday









md2perpe

7,72111028




7,72111028










asked yesterday









Math Lady

1146




1146








  • 3




    I'm trying to understand your thinking when you went from $lim_{ntoinfty}$ to $frac{d}{dn}int_0^infty dn$. I don't see how that could work.
    – Teepeemm
    yesterday










  • (Another take on Teepeemm's comment:) $frac{mathrm{d}}{mathrm{d}n} langle text{expression containing no "$n$"s} rangle = 0$.
    – Eric Towers
    yesterday














  • 3




    I'm trying to understand your thinking when you went from $lim_{ntoinfty}$ to $frac{d}{dn}int_0^infty dn$. I don't see how that could work.
    – Teepeemm
    yesterday










  • (Another take on Teepeemm's comment:) $frac{mathrm{d}}{mathrm{d}n} langle text{expression containing no "$n$"s} rangle = 0$.
    – Eric Towers
    yesterday








3




3




I'm trying to understand your thinking when you went from $lim_{ntoinfty}$ to $frac{d}{dn}int_0^infty dn$. I don't see how that could work.
– Teepeemm
yesterday




I'm trying to understand your thinking when you went from $lim_{ntoinfty}$ to $frac{d}{dn}int_0^infty dn$. I don't see how that could work.
– Teepeemm
yesterday












(Another take on Teepeemm's comment:) $frac{mathrm{d}}{mathrm{d}n} langle text{expression containing no "$n$"s} rangle = 0$.
– Eric Towers
yesterday




(Another take on Teepeemm's comment:) $frac{mathrm{d}}{mathrm{d}n} langle text{expression containing no "$n$"s} rangle = 0$.
– Eric Towers
yesterday










2 Answers
2






active

oldest

votes


















7














Hint:



You have $$lim_n int_0^infty frac{x^n}{x^{n+3}+1} = lim_n int_0^1 frac{x^n}{x^{n+3}+1}+lim_n int_1^infty frac{x^n}{x^{n+3}+1}$$



Use on each term the dominated convergence theorem to get the limits inside. If $0<x<1$ then $x^nrightarrow 0$ so the first term is easy to calculate.



The second converges to $frac{1}{x^3}$ and then you need to evaluate $int_1^infty frac{1}{x^3} dx$.






share|cite|improve this answer





























    2














    Hint. Note that
    $$int_0^{infty}frac{x^n}{ x^{n+3}+1},dx=int_0^{1}frac{x^n}{ x^{n+3}+1},dx+int_1^{infty}frac{dx}{x^3}-int_1^{infty}frac{dx}{x^3( x^{n+3}+1)},$$
    where
    $$0leq int_0^1frac{x^n}{ x^{n+3}+1},dx leq int_0^1 x^n ,dx=left[frac{x^{n+1}}{n+1}right]_0^{1}=frac{1}{n+1},$$
    and
    $$0leq int_1^{infty}frac{dx}{x^3( x^{n+3}+1)}leq
    int_1^{infty}frac{1}{ x^{n+3}},dx=left[-frac{1}{(n+2)x^{n+2}}right]_1^{infty}=frac{1}{n+2}.$$






    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',
      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%2f3062748%2fmeasure-theory-integral%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









      7














      Hint:



      You have $$lim_n int_0^infty frac{x^n}{x^{n+3}+1} = lim_n int_0^1 frac{x^n}{x^{n+3}+1}+lim_n int_1^infty frac{x^n}{x^{n+3}+1}$$



      Use on each term the dominated convergence theorem to get the limits inside. If $0<x<1$ then $x^nrightarrow 0$ so the first term is easy to calculate.



      The second converges to $frac{1}{x^3}$ and then you need to evaluate $int_1^infty frac{1}{x^3} dx$.






      share|cite|improve this answer


























        7














        Hint:



        You have $$lim_n int_0^infty frac{x^n}{x^{n+3}+1} = lim_n int_0^1 frac{x^n}{x^{n+3}+1}+lim_n int_1^infty frac{x^n}{x^{n+3}+1}$$



        Use on each term the dominated convergence theorem to get the limits inside. If $0<x<1$ then $x^nrightarrow 0$ so the first term is easy to calculate.



        The second converges to $frac{1}{x^3}$ and then you need to evaluate $int_1^infty frac{1}{x^3} dx$.






        share|cite|improve this answer
























          7












          7








          7






          Hint:



          You have $$lim_n int_0^infty frac{x^n}{x^{n+3}+1} = lim_n int_0^1 frac{x^n}{x^{n+3}+1}+lim_n int_1^infty frac{x^n}{x^{n+3}+1}$$



          Use on each term the dominated convergence theorem to get the limits inside. If $0<x<1$ then $x^nrightarrow 0$ so the first term is easy to calculate.



          The second converges to $frac{1}{x^3}$ and then you need to evaluate $int_1^infty frac{1}{x^3} dx$.






          share|cite|improve this answer












          Hint:



          You have $$lim_n int_0^infty frac{x^n}{x^{n+3}+1} = lim_n int_0^1 frac{x^n}{x^{n+3}+1}+lim_n int_1^infty frac{x^n}{x^{n+3}+1}$$



          Use on each term the dominated convergence theorem to get the limits inside. If $0<x<1$ then $x^nrightarrow 0$ so the first term is easy to calculate.



          The second converges to $frac{1}{x^3}$ and then you need to evaluate $int_1^infty frac{1}{x^3} dx$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered yesterday









          Yanko

          6,394727




          6,394727























              2














              Hint. Note that
              $$int_0^{infty}frac{x^n}{ x^{n+3}+1},dx=int_0^{1}frac{x^n}{ x^{n+3}+1},dx+int_1^{infty}frac{dx}{x^3}-int_1^{infty}frac{dx}{x^3( x^{n+3}+1)},$$
              where
              $$0leq int_0^1frac{x^n}{ x^{n+3}+1},dx leq int_0^1 x^n ,dx=left[frac{x^{n+1}}{n+1}right]_0^{1}=frac{1}{n+1},$$
              and
              $$0leq int_1^{infty}frac{dx}{x^3( x^{n+3}+1)}leq
              int_1^{infty}frac{1}{ x^{n+3}},dx=left[-frac{1}{(n+2)x^{n+2}}right]_1^{infty}=frac{1}{n+2}.$$






              share|cite|improve this answer




























                2














                Hint. Note that
                $$int_0^{infty}frac{x^n}{ x^{n+3}+1},dx=int_0^{1}frac{x^n}{ x^{n+3}+1},dx+int_1^{infty}frac{dx}{x^3}-int_1^{infty}frac{dx}{x^3( x^{n+3}+1)},$$
                where
                $$0leq int_0^1frac{x^n}{ x^{n+3}+1},dx leq int_0^1 x^n ,dx=left[frac{x^{n+1}}{n+1}right]_0^{1}=frac{1}{n+1},$$
                and
                $$0leq int_1^{infty}frac{dx}{x^3( x^{n+3}+1)}leq
                int_1^{infty}frac{1}{ x^{n+3}},dx=left[-frac{1}{(n+2)x^{n+2}}right]_1^{infty}=frac{1}{n+2}.$$






                share|cite|improve this answer


























                  2












                  2








                  2






                  Hint. Note that
                  $$int_0^{infty}frac{x^n}{ x^{n+3}+1},dx=int_0^{1}frac{x^n}{ x^{n+3}+1},dx+int_1^{infty}frac{dx}{x^3}-int_1^{infty}frac{dx}{x^3( x^{n+3}+1)},$$
                  where
                  $$0leq int_0^1frac{x^n}{ x^{n+3}+1},dx leq int_0^1 x^n ,dx=left[frac{x^{n+1}}{n+1}right]_0^{1}=frac{1}{n+1},$$
                  and
                  $$0leq int_1^{infty}frac{dx}{x^3( x^{n+3}+1)}leq
                  int_1^{infty}frac{1}{ x^{n+3}},dx=left[-frac{1}{(n+2)x^{n+2}}right]_1^{infty}=frac{1}{n+2}.$$






                  share|cite|improve this answer














                  Hint. Note that
                  $$int_0^{infty}frac{x^n}{ x^{n+3}+1},dx=int_0^{1}frac{x^n}{ x^{n+3}+1},dx+int_1^{infty}frac{dx}{x^3}-int_1^{infty}frac{dx}{x^3( x^{n+3}+1)},$$
                  where
                  $$0leq int_0^1frac{x^n}{ x^{n+3}+1},dx leq int_0^1 x^n ,dx=left[frac{x^{n+1}}{n+1}right]_0^{1}=frac{1}{n+1},$$
                  and
                  $$0leq int_1^{infty}frac{dx}{x^3( x^{n+3}+1)}leq
                  int_1^{infty}frac{1}{ x^{n+3}},dx=left[-frac{1}{(n+2)x^{n+2}}right]_1^{infty}=frac{1}{n+2}.$$







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited yesterday

























                  answered yesterday









                  Robert Z

                  93.7k1061132




                  93.7k1061132






























                      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.





                      Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


                      Please pay close attention to the following guidance:


                      • 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.


                      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%2f3062748%2fmeasure-theory-integral%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 make a Squid Proxy server?

                      Image
                      1 I want to set up squid proxy server for monitoring user activity in a network and Internet Web URL filtering for user. I also want to configure the user login, user log, user access group, and block unwanted web sites in my network. Please help me and guide me the full process to do this. server proxy squid share | improve this question edited Jul 31 '12 at 14:45 Jorge Castro 37.1k 107 422 617 asked Jul 31 '12 at 5:16 Goivind Tiwari Goivind Tiwari ...
                      Read more

                      Is this a new Fibonacci Identity?

                      Image
                      6 1 $begingroup$ I have found the following Fibonacci Identity (and proved it). If $F_n$ denotes the nth Fibonacci Number, we have the following identity begin{equation} F_{n-r+h}F_{n+k+g+1} - F_{n-r+g}F_{n+k+h+1} = (-1)^{n+r+h+1} F_{g-h}F_{k+r+1} end{equation} where $F_1 = F_2 = 1$ , $r leq n$ , $h leq g$ , and $n, g, k in mathbb{N}$ . It is not too hard to show that this identity subsumes Cassini's Identity, Catalan's Identity, Vajda's Idenity, and d'Ocagne's identity to name a few. I have done a pretty thorough literature review, and I have not found anything like this, but I am still wondering if anyone has seen this identity before? I found this by accident after noticing some patterns in some analysis work I was doing, so if this is already known I would be curious to see w...
                      Read more

                      19世紀

                      Image
                      この記事は検証可能な参考文献や出典が全く示されていないか、不十分です。 出典を追加して記事の信頼性向上にご協力ください。 （ 2012年3月 ） 千年紀: 2千年紀 世紀: 18世紀 - 19世紀 - 20世紀 十年紀: 1800年代 1810年代 1820年代 1830年代 1840年代 1850年代 1860年代 1870年代 1880年代 1890年代 19世紀に君臨した大英帝国。 19世紀 （じゅうきゅうせいき）は、西暦1801年から西暦1900年までの100年間を指す世紀。 目次 1 19世紀の歴史 1.1 国民国家の成立 1.2 帝国主義の興隆 1.3 列強の植民地争奪戦 2 できごと 2.1 1800年代 2.2 1810年代 2.3 1820年代 2.4 1830年代 2.5 1840年代 2.6 1850年代 2.7 1860年代 2.8 1870年代 2.9 1880年代 2.10 1890年代 2.11 1900年代 3 文化 3.1 文学 3.2 音楽 3.3 思想 3.4 科学 3.5 技術 4 人物 4.1 ヨーロッパ 4.1.1 政治家・王族 4.1.1.1 フランス 4.1.1.2 オーストリア＝ハンガリー 4.1.1.3 ロシア 4.1.1.4 イギリス 4.1.1.5 ドイツ（プロイセンほかドイツ領邦を含む） 4.1.1.6 北欧 4.1.1.7 イタリア 4.1.1.8 スペイン 4.1.1.9 ベルギー 4.1.1.10 ギリシア 4.1.2 軍人 4.1.3 実業家 4.1.4 科学と技術 4.1.5 思想と哲学・人文諸学 4.1.6 宗教 4.1.7 文学 4.1.8 美術 4.1.9 音楽 4.1.10 社会事業家 4.1.11 探検家・旅行家 4.1.12 料...
                      Read more