Limit of terms of Harmonic series












1












$begingroup$


Let $s(n)$ be the smallest positive integer such that $1+frac{1}{2}+frac{1}{3}+...+1/s(n)geq{n}$



Find $displaystylelim_{ntoinfty} frac{s(n+1)}{s(n)}$



The values of $s(n)$ are



$s(1)=1, s(2)=4, s(3)=11, s(4)=31, s(5)=83, s(6)=227, s(7)=616 $ and so on



$frac{s(2)}{s(1)}=4, frac{s(3)}{s(2)}=2.75, frac{s(4)}{s(3)}=2.8181..., frac{s(5)}{s(4)}=2.677..., frac{s(6)}{s(5)}=2.73493..., frac{s(7)}{s(6)}=2.713...,$



I currently can't find a formula for $s(n)$ but from the calculations I can see that the ratio converges to $e$.



Any tips on how to prove this?






sequences-and-series







share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    Let $s(n)$ be the smallest positive integer such that $1+frac{1}{2}+frac{1}{3}+...+1/s(n)geq{n}$



    Find $displaystylelim_{ntoinfty} frac{s(n+1)}{s(n)}$



    The values of $s(n)$ are



    $s(1)=1, s(2)=4, s(3)=11, s(4)=31, s(5)=83, s(6)=227, s(7)=616 $ and so on



    $frac{s(2)}{s(1)}=4, frac{s(3)}{s(2)}=2.75, frac{s(4)}{s(3)}=2.8181..., frac{s(5)}{s(4)}=2.677..., frac{s(6)}{s(5)}=2.73493..., frac{s(7)}{s(6)}=2.713...,$



    I currently can't find a formula for $s(n)$ but from the calculations I can see that the ratio converges to $e$.



    Any tips on how to prove this?






    sequences-and-series







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      Let $s(n)$ be the smallest positive integer such that $1+frac{1}{2}+frac{1}{3}+...+1/s(n)geq{n}$



      Find $displaystylelim_{ntoinfty} frac{s(n+1)}{s(n)}$



      The values of $s(n)$ are



      $s(1)=1, s(2)=4, s(3)=11, s(4)=31, s(5)=83, s(6)=227, s(7)=616 $ and so on



      $frac{s(2)}{s(1)}=4, frac{s(3)}{s(2)}=2.75, frac{s(4)}{s(3)}=2.8181..., frac{s(5)}{s(4)}=2.677..., frac{s(6)}{s(5)}=2.73493..., frac{s(7)}{s(6)}=2.713...,$



      I currently can't find a formula for $s(n)$ but from the calculations I can see that the ratio converges to $e$.



      Any tips on how to prove this?






      sequences-and-series







      share|cite|improve this question









      $endgroup$




      Let $s(n)$ be the smallest positive integer such that $1+frac{1}{2}+frac{1}{3}+...+1/s(n)geq{n}$



      Find $displaystylelim_{ntoinfty} frac{s(n+1)}{s(n)}$



      The values of $s(n)$ are



      $s(1)=1, s(2)=4, s(3)=11, s(4)=31, s(5)=83, s(6)=227, s(7)=616 $ and so on



      $frac{s(2)}{s(1)}=4, frac{s(3)}{s(2)}=2.75, frac{s(4)}{s(3)}=2.8181..., frac{s(5)}{s(4)}=2.677..., frac{s(6)}{s(5)}=2.73493..., frac{s(7)}{s(6)}=2.713...,$



      I currently can't find a formula for $s(n)$ but from the calculations I can see that the ratio converges to $e$.



      Any tips on how to prove this?






      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










      asked Dec 6 '18 at 14:16









      A. BrunoA. Bruno

      1035




      1035






















          1 Answer
          1






          active

          oldest

          votes


















          0












          $begingroup$

          The point is that the partial sums of the harmonic series are slowly increasing:
          $$ H_n = sum_{k=1}^{n}frac{1}{k} = log n + underbrace{sum_{kgeq 1}left[frac{1}{n}-logleft(1+frac{1}{n}right)right]}_{gamma,approxfrac{1}{sqrt{3}}}+Oleft(frac{1}{n}right) $$
          hence for any $varepsilon>0$ and for any $N$ large enough, the smallest $n$ such that $H_ngeq N$ is bounded between $expleft(N-gamma-varepsilonright)$ and $exp(N-gamma+varepsilon)$. In particular the wanted limit equals $exp(1)=e$.






          share|cite|improve this answer









          $endgroup$













            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%2f3028540%2flimit-of-terms-of-harmonic-series%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            0












            $begingroup$

            The point is that the partial sums of the harmonic series are slowly increasing:
            $$ H_n = sum_{k=1}^{n}frac{1}{k} = log n + underbrace{sum_{kgeq 1}left[frac{1}{n}-logleft(1+frac{1}{n}right)right]}_{gamma,approxfrac{1}{sqrt{3}}}+Oleft(frac{1}{n}right) $$
            hence for any $varepsilon>0$ and for any $N$ large enough, the smallest $n$ such that $H_ngeq N$ is bounded between $expleft(N-gamma-varepsilonright)$ and $exp(N-gamma+varepsilon)$. In particular the wanted limit equals $exp(1)=e$.






            share|cite|improve this answer









            $endgroup$


















              0












              $begingroup$

              The point is that the partial sums of the harmonic series are slowly increasing:
              $$ H_n = sum_{k=1}^{n}frac{1}{k} = log n + underbrace{sum_{kgeq 1}left[frac{1}{n}-logleft(1+frac{1}{n}right)right]}_{gamma,approxfrac{1}{sqrt{3}}}+Oleft(frac{1}{n}right) $$
              hence for any $varepsilon>0$ and for any $N$ large enough, the smallest $n$ such that $H_ngeq N$ is bounded between $expleft(N-gamma-varepsilonright)$ and $exp(N-gamma+varepsilon)$. In particular the wanted limit equals $exp(1)=e$.






              share|cite|improve this answer









              $endgroup$
















                0












                0








                0





                $begingroup$

                The point is that the partial sums of the harmonic series are slowly increasing:
                $$ H_n = sum_{k=1}^{n}frac{1}{k} = log n + underbrace{sum_{kgeq 1}left[frac{1}{n}-logleft(1+frac{1}{n}right)right]}_{gamma,approxfrac{1}{sqrt{3}}}+Oleft(frac{1}{n}right) $$
                hence for any $varepsilon>0$ and for any $N$ large enough, the smallest $n$ such that $H_ngeq N$ is bounded between $expleft(N-gamma-varepsilonright)$ and $exp(N-gamma+varepsilon)$. In particular the wanted limit equals $exp(1)=e$.






                share|cite|improve this answer









                $endgroup$



                The point is that the partial sums of the harmonic series are slowly increasing:
                $$ H_n = sum_{k=1}^{n}frac{1}{k} = log n + underbrace{sum_{kgeq 1}left[frac{1}{n}-logleft(1+frac{1}{n}right)right]}_{gamma,approxfrac{1}{sqrt{3}}}+Oleft(frac{1}{n}right) $$
                hence for any $varepsilon>0$ and for any $N$ large enough, the smallest $n$ such that $H_ngeq N$ is bounded between $expleft(N-gamma-varepsilonright)$ and $exp(N-gamma+varepsilon)$. In particular the wanted limit equals $exp(1)=e$.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 6 '18 at 14:31









                Jack D'AurizioJack D'Aurizio

                288k33280659




                288k33280659






























                    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%2f3028540%2flimit-of-terms-of-harmonic-series%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

                    To store a contact into the json file from server.js file using a class in NodeJS

                    Image
                    0 I have created three files -contactPresentation.js, contactService.js, contacts.json. I will run the contactService.js file. And it will navigate to contactPresentation.js file that will only choose an option to either add or display a file. If add(), then add the data into add of service file.And the data will be added to the contacts.json file . And if display(), then display the display() of service file. And the display will take the data from contacts.json file. However, i know my code is flawed. Please correct it . I am also getting an error - this.contacts.push(new ContactService(userName, contactNumber, emailId)); ^ TypeError: Cannot read property 'push' of undefined at ContactService.add (e:NodeJSProjectcontactManager2contactService.js:57:23) at new C
                    Read more

                    Redirect URL with Chrome Remote Debugging Android Devices

                    Image
                    1 How can i redirect an URL with Chrome Remote Debugging Android Devices? like Requestly Chrome extension on desktop. google-chrome debugging url redirect testing share | improve this question edited Nov 24 '18 at 8:16 sachinjain024 14.7k 23 73 129 asked Nov 23 '18 at 16:09 Momen Momen 91 10
                    Read more

                    Dieringhausen

                    Image
                    Dieringhausen Stadt Gummersbach 50.982777777778 7.5316666666667 165 Koordinaten: 50° 58′ 58″  N , 7° 31′ 54″  O Höhe: 165  (160–245)  m Einwohner: 5055  (30. Jun. 2016) Postleitzahl: 51645 Vorwahl: 02261 Lage von Dieringhausen in Gummersbach Im Hohl ca. 1927–1930 Ansicht von Südwesten bei Bünghausen Dieringhausen (im örtlichen Dialekt Dierkesen ) ist ein Ortsteil von Gummersbach im Oberbergischen Kreis, Regierungsbezirk Köln, Nordrhein-Westfalen (Deutschland). Inhaltsverzeichnis 1 Geographie 2 Geschichte 3 Kultur 3.1 Sehenswürdigkeiten 3.2 Regelmäßige Veranstaltungen 4 Infrastruktur und Wirtschaft 4.1 Verkehr 4.1.1 Schienen- und Busverkehr 4.1.2 Straßen 4.2 Öffentliche Einrichtungen 4.2.1 Schulen und Bildungseinrichtungen 4.2.2 Kirchliche Einrichtungen 5 Persönlichkeiten 5.1 In Dieringhausen geboren 5.2 In Dieringhausen gelebt 6 Einzelnachweise 7
                    Read more