How to find the Limit.











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Given the following question first I thought I could use Caesaro's result somehow to solve this but it went nowhere.
enter image description here



Kindly help with this problem I am missing the correct approach.



Thanks and regards in advance






real-analysis sequences-and-series







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    up vote
    0
    down vote

    favorite












    Given the following question first I thought I could use Caesaro's result somehow to solve this but it went nowhere.
    enter image description here



    Kindly help with this problem I am missing the correct approach.



    Thanks and regards in advance






    real-analysis sequences-and-series







    share|cite|improve this question
























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Given the following question first I thought I could use Caesaro's result somehow to solve this but it went nowhere.
      enter image description here



      Kindly help with this problem I am missing the correct approach.



      Thanks and regards in advance






      real-analysis sequences-and-series







      share|cite|improve this question













      Given the following question first I thought I could use Caesaro's result somehow to solve this but it went nowhere.
      enter image description here



      Kindly help with this problem I am missing the correct approach.



      Thanks and regards in advance






      real-analysis sequences-and-series




      real-analysis sequences-and-series






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Nov 15 at 16:41









      Devendra Singh Rana

      744216




      744216






















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          HINT



          By Stolz–Cesàro theorem we have



          $$lim_{nto infty} frac{sum_{i=1}^n a_ix_i}{s_n}=lim_{nto infty} frac{ sum_{i=1}^{n+1} a_ix_i-sum_{i=1}^n a_ix_i}{s_{n+1}-s_n}$$



          when the RHS exists.






          share|cite|improve this answer























          • I am not aware of this theorem@gimusi any links will be helpful.
            – Devendra Singh Rana
            Nov 15 at 17:17






          • 1




            @DevendraSinghRana I've added a reference for that theorem.
            – gimusi
            Nov 15 at 17:22











          Your Answer





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          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          2
          down vote



          accepted










          HINT



          By Stolz–Cesàro theorem we have



          $$lim_{nto infty} frac{sum_{i=1}^n a_ix_i}{s_n}=lim_{nto infty} frac{ sum_{i=1}^{n+1} a_ix_i-sum_{i=1}^n a_ix_i}{s_{n+1}-s_n}$$



          when the RHS exists.






          share|cite|improve this answer























          • I am not aware of this theorem@gimusi any links will be helpful.
            – Devendra Singh Rana
            Nov 15 at 17:17






          • 1




            @DevendraSinghRana I've added a reference for that theorem.
            – gimusi
            Nov 15 at 17:22















          up vote
          2
          down vote



          accepted










          HINT



          By Stolz–Cesàro theorem we have



          $$lim_{nto infty} frac{sum_{i=1}^n a_ix_i}{s_n}=lim_{nto infty} frac{ sum_{i=1}^{n+1} a_ix_i-sum_{i=1}^n a_ix_i}{s_{n+1}-s_n}$$



          when the RHS exists.






          share|cite|improve this answer























          • I am not aware of this theorem@gimusi any links will be helpful.
            – Devendra Singh Rana
            Nov 15 at 17:17






          • 1




            @DevendraSinghRana I've added a reference for that theorem.
            – gimusi
            Nov 15 at 17:22













          up vote
          2
          down vote



          accepted







          up vote
          2
          down vote



          accepted






          HINT



          By Stolz–Cesàro theorem we have



          $$lim_{nto infty} frac{sum_{i=1}^n a_ix_i}{s_n}=lim_{nto infty} frac{ sum_{i=1}^{n+1} a_ix_i-sum_{i=1}^n a_ix_i}{s_{n+1}-s_n}$$



          when the RHS exists.






          share|cite|improve this answer














          HINT



          By Stolz–Cesàro theorem we have



          $$lim_{nto infty} frac{sum_{i=1}^n a_ix_i}{s_n}=lim_{nto infty} frac{ sum_{i=1}^{n+1} a_ix_i-sum_{i=1}^n a_ix_i}{s_{n+1}-s_n}$$



          when the RHS exists.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Nov 15 at 17:21

























          answered Nov 15 at 17:12









          gimusi

          87.2k74393




          87.2k74393












          • I am not aware of this theorem@gimusi any links will be helpful.
            – Devendra Singh Rana
            Nov 15 at 17:17






          • 1




            @DevendraSinghRana I've added a reference for that theorem.
            – gimusi
            Nov 15 at 17:22


















          • I am not aware of this theorem@gimusi any links will be helpful.
            – Devendra Singh Rana
            Nov 15 at 17:17






          • 1




            @DevendraSinghRana I've added a reference for that theorem.
            – gimusi
            Nov 15 at 17:22
















          I am not aware of this theorem@gimusi any links will be helpful.
          – Devendra Singh Rana
          Nov 15 at 17:17




          I am not aware of this theorem@gimusi any links will be helpful.
          – Devendra Singh Rana
          Nov 15 at 17:17




          1




          1




          @DevendraSinghRana I've added a reference for that theorem.
          – gimusi
          Nov 15 at 17:22




          @DevendraSinghRana I've added a reference for that theorem.
          – gimusi
          Nov 15 at 17:22


















           

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