Infinite linear combination of linearly independent reals equaling 0

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
1
down vote

favorite












Is it possible to find a sequence of real numbers $x_n$ linearly independent over $mathbbQ$ with the property that there exists a sequence of rationals $q_n$ such that $sum q_n x_n = 0,$ but for any sequence of integers $a_n, sum a_nx_n not= 0$?







share|cite|improve this question
























    up vote
    1
    down vote

    favorite












    Is it possible to find a sequence of real numbers $x_n$ linearly independent over $mathbbQ$ with the property that there exists a sequence of rationals $q_n$ such that $sum q_n x_n = 0,$ but for any sequence of integers $a_n, sum a_nx_n not= 0$?







    share|cite|improve this question






















      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      Is it possible to find a sequence of real numbers $x_n$ linearly independent over $mathbbQ$ with the property that there exists a sequence of rationals $q_n$ such that $sum q_n x_n = 0,$ but for any sequence of integers $a_n, sum a_nx_n not= 0$?







      share|cite|improve this question












      Is it possible to find a sequence of real numbers $x_n$ linearly independent over $mathbbQ$ with the property that there exists a sequence of rationals $q_n$ such that $sum q_n x_n = 0,$ but for any sequence of integers $a_n, sum a_nx_n not= 0$?









      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Aug 25 at 5:34









      cats

      3,443726




      3,443726




















          1 Answer
          1






          active

          oldest

          votes

















          up vote
          1
          down vote



          accepted










          Of course; take any transcendental number $0<alpha<1$ and set $x_n:=n(alpha^n+1-alpha^n)$ and $q_n=frac1n$.






          share|cite|improve this answer




















          • Can you elaborate on why this implies there do not exist integers satisfying the same condition?
            – cats
            Aug 25 at 15:59










          • It turns out this question is quite easy, though I do not believe the above sequence works since the limit of $x_n$ is $0.$ Any sequence with lim inf not $0$ will do the trick.
            – cats
            Aug 26 at 5:51










          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: false,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          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%2f2893818%2finfinite-linear-combination-of-linearly-independent-reals-equaling-0%23new-answer', 'question_page');

          );

          Post as a guest






























          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          1
          down vote



          accepted










          Of course; take any transcendental number $0<alpha<1$ and set $x_n:=n(alpha^n+1-alpha^n)$ and $q_n=frac1n$.






          share|cite|improve this answer




















          • Can you elaborate on why this implies there do not exist integers satisfying the same condition?
            – cats
            Aug 25 at 15:59










          • It turns out this question is quite easy, though I do not believe the above sequence works since the limit of $x_n$ is $0.$ Any sequence with lim inf not $0$ will do the trick.
            – cats
            Aug 26 at 5:51














          up vote
          1
          down vote



          accepted










          Of course; take any transcendental number $0<alpha<1$ and set $x_n:=n(alpha^n+1-alpha^n)$ and $q_n=frac1n$.






          share|cite|improve this answer




















          • Can you elaborate on why this implies there do not exist integers satisfying the same condition?
            – cats
            Aug 25 at 15:59










          • It turns out this question is quite easy, though I do not believe the above sequence works since the limit of $x_n$ is $0.$ Any sequence with lim inf not $0$ will do the trick.
            – cats
            Aug 26 at 5:51












          up vote
          1
          down vote



          accepted







          up vote
          1
          down vote



          accepted






          Of course; take any transcendental number $0<alpha<1$ and set $x_n:=n(alpha^n+1-alpha^n)$ and $q_n=frac1n$.






          share|cite|improve this answer












          Of course; take any transcendental number $0<alpha<1$ and set $x_n:=n(alpha^n+1-alpha^n)$ and $q_n=frac1n$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Aug 25 at 9:37









          Servaes

          1




          1











          • Can you elaborate on why this implies there do not exist integers satisfying the same condition?
            – cats
            Aug 25 at 15:59










          • It turns out this question is quite easy, though I do not believe the above sequence works since the limit of $x_n$ is $0.$ Any sequence with lim inf not $0$ will do the trick.
            – cats
            Aug 26 at 5:51
















          • Can you elaborate on why this implies there do not exist integers satisfying the same condition?
            – cats
            Aug 25 at 15:59










          • It turns out this question is quite easy, though I do not believe the above sequence works since the limit of $x_n$ is $0.$ Any sequence with lim inf not $0$ will do the trick.
            – cats
            Aug 26 at 5:51















          Can you elaborate on why this implies there do not exist integers satisfying the same condition?
          – cats
          Aug 25 at 15:59




          Can you elaborate on why this implies there do not exist integers satisfying the same condition?
          – cats
          Aug 25 at 15:59












          It turns out this question is quite easy, though I do not believe the above sequence works since the limit of $x_n$ is $0.$ Any sequence with lim inf not $0$ will do the trick.
          – cats
          Aug 26 at 5:51




          It turns out this question is quite easy, though I do not believe the above sequence works since the limit of $x_n$ is $0.$ Any sequence with lim inf not $0$ will do the trick.
          – cats
          Aug 26 at 5:51

















           

          draft saved


          draft discarded















































           


          draft saved


          draft discarded














          StackExchange.ready(
          function ()
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2893818%2finfinite-linear-combination-of-linearly-independent-reals-equaling-0%23new-answer', 'question_page');

          );

          Post as a guest













































































          這個網誌中的熱門文章

          How to combine Bézier curves to a surface?

          Carbon dioxide

          Why am i infinitely getting the same tweet with the Twitter Search API?