Determine whether the sequence $sinfrac{npi}{2}$ converges and prove your answer using the epsilon method..












1














I am having the most trouble with the scratch work ahead of the actual proof. I can't figure out proof of $x_n=sinfrac{npi}{2}$ beyond $epsilon in mathbb{R}$ and $epsilon > 0$.










share|cite|improve this question
























  • Have you tried computing $x_n$ for small $n$, say $0 leq n leq 5$?
    – JavaMan
    Nov 29 '18 at 3:44






  • 1




    The subsequence $x_{2n}equiv 0$. The subsequence $x_{4n+1}=1$. The subsequence $x_{4n+3}=-1$. Conclude.
    – Mark Viola
    Nov 29 '18 at 3:47


















1














I am having the most trouble with the scratch work ahead of the actual proof. I can't figure out proof of $x_n=sinfrac{npi}{2}$ beyond $epsilon in mathbb{R}$ and $epsilon > 0$.










share|cite|improve this question
























  • Have you tried computing $x_n$ for small $n$, say $0 leq n leq 5$?
    – JavaMan
    Nov 29 '18 at 3:44






  • 1




    The subsequence $x_{2n}equiv 0$. The subsequence $x_{4n+1}=1$. The subsequence $x_{4n+3}=-1$. Conclude.
    – Mark Viola
    Nov 29 '18 at 3:47
















1












1








1







I am having the most trouble with the scratch work ahead of the actual proof. I can't figure out proof of $x_n=sinfrac{npi}{2}$ beyond $epsilon in mathbb{R}$ and $epsilon > 0$.










share|cite|improve this question















I am having the most trouble with the scratch work ahead of the actual proof. I can't figure out proof of $x_n=sinfrac{npi}{2}$ beyond $epsilon in mathbb{R}$ and $epsilon > 0$.







sequences-and-series limits epsilon-delta






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 29 '18 at 4:24









Key Flex

7,75741232




7,75741232










asked Nov 29 '18 at 3:40









NeedHelpNeedHelp

161




161












  • Have you tried computing $x_n$ for small $n$, say $0 leq n leq 5$?
    – JavaMan
    Nov 29 '18 at 3:44






  • 1




    The subsequence $x_{2n}equiv 0$. The subsequence $x_{4n+1}=1$. The subsequence $x_{4n+3}=-1$. Conclude.
    – Mark Viola
    Nov 29 '18 at 3:47




















  • Have you tried computing $x_n$ for small $n$, say $0 leq n leq 5$?
    – JavaMan
    Nov 29 '18 at 3:44






  • 1




    The subsequence $x_{2n}equiv 0$. The subsequence $x_{4n+1}=1$. The subsequence $x_{4n+3}=-1$. Conclude.
    – Mark Viola
    Nov 29 '18 at 3:47


















Have you tried computing $x_n$ for small $n$, say $0 leq n leq 5$?
– JavaMan
Nov 29 '18 at 3:44




Have you tried computing $x_n$ for small $n$, say $0 leq n leq 5$?
– JavaMan
Nov 29 '18 at 3:44




1




1




The subsequence $x_{2n}equiv 0$. The subsequence $x_{4n+1}=1$. The subsequence $x_{4n+3}=-1$. Conclude.
– Mark Viola
Nov 29 '18 at 3:47






The subsequence $x_{2n}equiv 0$. The subsequence $x_{4n+1}=1$. The subsequence $x_{4n+3}=-1$. Conclude.
– Mark Viola
Nov 29 '18 at 3:47












4 Answers
4






active

oldest

votes


















2














HINT:



Note that the subsequence $x_{2n}equiv 0$, the subsequence $x_{4n+1}equiv1$, and the subsequence $x_{4n+3}equiv-1$.



What can you conclude now?






share|cite|improve this answer





























    1














    It might be helpful to consider the actual values of $sin(npi / 2)$ where $n in mathbb{N}$ to obtain an alternate definition of the sequence.



    For example, try $n = 1, 2, 3, 4,...$ and so on, and see if you can find a pattern. This seems more like a trick question because of that: once you get the other, equivalent sequence, things should be much easier.



    (You might have to properly justify that the two sequences are equivalent though but that should be easy enough.)






    share|cite|improve this answer





























      1














      If the series converges to some real number $l$, then $forallvarepsilon>0, exists n_0inmathbb N$ such that $|sin frac{npi}{2}-l|<varepsilon, forall ngeq n_0$. It suffices to show that $existsvarepsilon>0$ for which no such $n_0$ exists.



      Say $varepsilon=1/10$. The interval $(l-varepsilon, l+varepsilon)equiv(l-1/10, l+1/10)$ has length 1/5. Say $sin frac{npi}{2}in(l-1/10, l+1/10)$



      $sin frac{(n+2)pi}{2}=-sinfrac{npi}{2}in(l-1/10, l+1/10)$



      or, by the definition of an interval, $(-sinfrac{npi}{2}, sin frac{npi}{2})subset(l-1/10, l+1/10)$. Whatever be $n_0$ in this case, you can always find a number of the form $n=4m+3geq n_0, minmathbb N$.



      $(-sinfrac{npi}{2}, sin frac{npi}{2})equiv(-1,1)$



      What is the length of this interval?






      share|cite|improve this answer





























        0














        Let $$a_n=x_{2n}=sin npi =0\b_n=x_{{4n+1}}=sin {4n+1over 2}pi=1$$since $a_n$ and $b_n$ are convergent to different limits, therefore $x_n$ is not convergent.






        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%2f3018138%2fdetermine-whether-the-sequence-sin-fracn-pi2-converges-and-prove-your-ans%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown

























          4 Answers
          4






          active

          oldest

          votes








          4 Answers
          4






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          2














          HINT:



          Note that the subsequence $x_{2n}equiv 0$, the subsequence $x_{4n+1}equiv1$, and the subsequence $x_{4n+3}equiv-1$.



          What can you conclude now?






          share|cite|improve this answer


























            2














            HINT:



            Note that the subsequence $x_{2n}equiv 0$, the subsequence $x_{4n+1}equiv1$, and the subsequence $x_{4n+3}equiv-1$.



            What can you conclude now?






            share|cite|improve this answer
























              2












              2








              2






              HINT:



              Note that the subsequence $x_{2n}equiv 0$, the subsequence $x_{4n+1}equiv1$, and the subsequence $x_{4n+3}equiv-1$.



              What can you conclude now?






              share|cite|improve this answer












              HINT:



              Note that the subsequence $x_{2n}equiv 0$, the subsequence $x_{4n+1}equiv1$, and the subsequence $x_{4n+3}equiv-1$.



              What can you conclude now?







              share|cite|improve this answer












              share|cite|improve this answer



              share|cite|improve this answer










              answered Nov 29 '18 at 3:52









              Mark ViolaMark Viola

              130k1274170




              130k1274170























                  1














                  It might be helpful to consider the actual values of $sin(npi / 2)$ where $n in mathbb{N}$ to obtain an alternate definition of the sequence.



                  For example, try $n = 1, 2, 3, 4,...$ and so on, and see if you can find a pattern. This seems more like a trick question because of that: once you get the other, equivalent sequence, things should be much easier.



                  (You might have to properly justify that the two sequences are equivalent though but that should be easy enough.)






                  share|cite|improve this answer


























                    1














                    It might be helpful to consider the actual values of $sin(npi / 2)$ where $n in mathbb{N}$ to obtain an alternate definition of the sequence.



                    For example, try $n = 1, 2, 3, 4,...$ and so on, and see if you can find a pattern. This seems more like a trick question because of that: once you get the other, equivalent sequence, things should be much easier.



                    (You might have to properly justify that the two sequences are equivalent though but that should be easy enough.)






                    share|cite|improve this answer
























                      1












                      1








                      1






                      It might be helpful to consider the actual values of $sin(npi / 2)$ where $n in mathbb{N}$ to obtain an alternate definition of the sequence.



                      For example, try $n = 1, 2, 3, 4,...$ and so on, and see if you can find a pattern. This seems more like a trick question because of that: once you get the other, equivalent sequence, things should be much easier.



                      (You might have to properly justify that the two sequences are equivalent though but that should be easy enough.)






                      share|cite|improve this answer












                      It might be helpful to consider the actual values of $sin(npi / 2)$ where $n in mathbb{N}$ to obtain an alternate definition of the sequence.



                      For example, try $n = 1, 2, 3, 4,...$ and so on, and see if you can find a pattern. This seems more like a trick question because of that: once you get the other, equivalent sequence, things should be much easier.



                      (You might have to properly justify that the two sequences are equivalent though but that should be easy enough.)







                      share|cite|improve this answer












                      share|cite|improve this answer



                      share|cite|improve this answer










                      answered Nov 29 '18 at 3:44









                      Eevee TrainerEevee Trainer

                      5,1071734




                      5,1071734























                          1














                          If the series converges to some real number $l$, then $forallvarepsilon>0, exists n_0inmathbb N$ such that $|sin frac{npi}{2}-l|<varepsilon, forall ngeq n_0$. It suffices to show that $existsvarepsilon>0$ for which no such $n_0$ exists.



                          Say $varepsilon=1/10$. The interval $(l-varepsilon, l+varepsilon)equiv(l-1/10, l+1/10)$ has length 1/5. Say $sin frac{npi}{2}in(l-1/10, l+1/10)$



                          $sin frac{(n+2)pi}{2}=-sinfrac{npi}{2}in(l-1/10, l+1/10)$



                          or, by the definition of an interval, $(-sinfrac{npi}{2}, sin frac{npi}{2})subset(l-1/10, l+1/10)$. Whatever be $n_0$ in this case, you can always find a number of the form $n=4m+3geq n_0, minmathbb N$.



                          $(-sinfrac{npi}{2}, sin frac{npi}{2})equiv(-1,1)$



                          What is the length of this interval?






                          share|cite|improve this answer


























                            1














                            If the series converges to some real number $l$, then $forallvarepsilon>0, exists n_0inmathbb N$ such that $|sin frac{npi}{2}-l|<varepsilon, forall ngeq n_0$. It suffices to show that $existsvarepsilon>0$ for which no such $n_0$ exists.



                            Say $varepsilon=1/10$. The interval $(l-varepsilon, l+varepsilon)equiv(l-1/10, l+1/10)$ has length 1/5. Say $sin frac{npi}{2}in(l-1/10, l+1/10)$



                            $sin frac{(n+2)pi}{2}=-sinfrac{npi}{2}in(l-1/10, l+1/10)$



                            or, by the definition of an interval, $(-sinfrac{npi}{2}, sin frac{npi}{2})subset(l-1/10, l+1/10)$. Whatever be $n_0$ in this case, you can always find a number of the form $n=4m+3geq n_0, minmathbb N$.



                            $(-sinfrac{npi}{2}, sin frac{npi}{2})equiv(-1,1)$



                            What is the length of this interval?






                            share|cite|improve this answer
























                              1












                              1








                              1






                              If the series converges to some real number $l$, then $forallvarepsilon>0, exists n_0inmathbb N$ such that $|sin frac{npi}{2}-l|<varepsilon, forall ngeq n_0$. It suffices to show that $existsvarepsilon>0$ for which no such $n_0$ exists.



                              Say $varepsilon=1/10$. The interval $(l-varepsilon, l+varepsilon)equiv(l-1/10, l+1/10)$ has length 1/5. Say $sin frac{npi}{2}in(l-1/10, l+1/10)$



                              $sin frac{(n+2)pi}{2}=-sinfrac{npi}{2}in(l-1/10, l+1/10)$



                              or, by the definition of an interval, $(-sinfrac{npi}{2}, sin frac{npi}{2})subset(l-1/10, l+1/10)$. Whatever be $n_0$ in this case, you can always find a number of the form $n=4m+3geq n_0, minmathbb N$.



                              $(-sinfrac{npi}{2}, sin frac{npi}{2})equiv(-1,1)$



                              What is the length of this interval?






                              share|cite|improve this answer












                              If the series converges to some real number $l$, then $forallvarepsilon>0, exists n_0inmathbb N$ such that $|sin frac{npi}{2}-l|<varepsilon, forall ngeq n_0$. It suffices to show that $existsvarepsilon>0$ for which no such $n_0$ exists.



                              Say $varepsilon=1/10$. The interval $(l-varepsilon, l+varepsilon)equiv(l-1/10, l+1/10)$ has length 1/5. Say $sin frac{npi}{2}in(l-1/10, l+1/10)$



                              $sin frac{(n+2)pi}{2}=-sinfrac{npi}{2}in(l-1/10, l+1/10)$



                              or, by the definition of an interval, $(-sinfrac{npi}{2}, sin frac{npi}{2})subset(l-1/10, l+1/10)$. Whatever be $n_0$ in this case, you can always find a number of the form $n=4m+3geq n_0, minmathbb N$.



                              $(-sinfrac{npi}{2}, sin frac{npi}{2})equiv(-1,1)$



                              What is the length of this interval?







                              share|cite|improve this answer












                              share|cite|improve this answer



                              share|cite|improve this answer










                              answered Nov 29 '18 at 4:17









                              Shubham JohriShubham Johri

                              4,479717




                              4,479717























                                  0














                                  Let $$a_n=x_{2n}=sin npi =0\b_n=x_{{4n+1}}=sin {4n+1over 2}pi=1$$since $a_n$ and $b_n$ are convergent to different limits, therefore $x_n$ is not convergent.






                                  share|cite|improve this answer


























                                    0














                                    Let $$a_n=x_{2n}=sin npi =0\b_n=x_{{4n+1}}=sin {4n+1over 2}pi=1$$since $a_n$ and $b_n$ are convergent to different limits, therefore $x_n$ is not convergent.






                                    share|cite|improve this answer
























                                      0












                                      0








                                      0






                                      Let $$a_n=x_{2n}=sin npi =0\b_n=x_{{4n+1}}=sin {4n+1over 2}pi=1$$since $a_n$ and $b_n$ are convergent to different limits, therefore $x_n$ is not convergent.






                                      share|cite|improve this answer












                                      Let $$a_n=x_{2n}=sin npi =0\b_n=x_{{4n+1}}=sin {4n+1over 2}pi=1$$since $a_n$ and $b_n$ are convergent to different limits, therefore $x_n$ is not convergent.







                                      share|cite|improve this answer












                                      share|cite|improve this answer



                                      share|cite|improve this answer










                                      answered Nov 29 '18 at 8:16









                                      Mostafa AyazMostafa Ayaz

                                      14.2k3937




                                      14.2k3937






























                                          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%2f3018138%2fdetermine-whether-the-sequence-sin-fracn-pi2-converges-and-prove-your-ans%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

                                          Ellipse (mathématiques)

                                          Quarter-circle Tiles

                                          Mont Emei