limit of a sequenceI know from my intuition that the sequence x n = ( 1 − 1 3 ) 2 ( 1 − 1 6 ) 2 ( 1 − 1 10 ) 2 ⋯ ⋯ ( 1 − 1 n ( n + 1 ) 2 ) 2 , n ≥ 2is convergent. But i don't know how to prove it.I almost try to apply every theorem I know (for eg ratio test ,monotone convergence theorem,...). Help me to prove this.Proof or idea is needed.Where does the sequence converge to?

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limit of a sequenceI know from my intuition that the sequence      x    n    =            (      1      −                                                                                  1                                                                                                          3                                          )        2              (      1      −                                                                                  1                                                                                                          6                                          )        2              (      1      −                                                                                  1                                                                                                          10                                          )        2    ⋯  ⋯            (      1      −                                                                                  1                                                                                                                                                                                                                                              n                                              (                        n                        +                        1                        )                                                                                                                                                                                                                2                                                                                                              )        2    ,    n  ≥  2is convergent. But i don&#039;t know how to prove it.I almost try to apply every theorem I know (for eg ratio test ,monotone convergence theorem,...). Help me to prove this.Proof or idea is needed.Where does the sequence converge to?

Sean Sutton · Accepted Answer

∏                      k            =            2                    ∞                          (          1          −                      2                                          k                2                            +              k                                )                                    =                  lim                      n            →            ∞                                    ∏                      k            =            2                    n                                                    (              k              −              1              )                                      (              k              +              2              )                                                          k                                      (              k              +              1              )                                                                        =                  lim                      n            →            ∞                                                              (              n              −              1              )              !                                                  (              n              +              2              )              !                              /                            3              !                                                          n              !                              /                            1              !                                                  (              n              +              1              )              !                              /                            2              !                                                                        =                  1          3                          lim                      n            →            ∞                                                n            +            2                    n                                                  =                  1          3                    Your product is just the square of this one.

Noe Cowan · Accepted Answer

∏          k      =      2        n        (    1    −          2              k        (        k        +        1        )              )    =      ∏          k      =      2        n              (      k      +      2      )      (      k      −      1      )              k      (      k      +      1      )        =            4      ⋅      1              2      ⋅      3        ⋅            5      ⋅      2              3      ⋅      4        ⋅            6      ⋅      3              4      ⋅      5        ⋅  .  .  .  ⋅            n      (      n      −      3      )              (      n      −      2      )      (      n      −      1      )        ⋅            (      n      +      1      )      (      n      −      2      )              (      n      −      1      )      n        ⋅            (      n      +      2      )      (      n      −      1      )              n      (      n      +      1      )        =            n      +      2              3      n

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