Find the limit of x n = ∏ j = 2 n ( 1 − 2 j ( j + 1 ) ) 2

Question

Find the limit of       x    n    =      ∏          j      =      2              n                  (      1      −              2                  j          (          j          +          1          )                    )        2

Falpo359 · Accepted Answer

∏                      j            =            2                    n                          (          1          −                      2                          j              (              j              +              1              )                                )                                    =                  ∏                      j            =            2                    n                          (                                    (              j              −              1              )              (              j              +              2              )                                      j              (              j              +              1              )                                )                                                  =                              ∏                          j              =              2                        n                                              j              −              1                        j                                                ∏                          j              =              2                        n                                              j              +              2                                      j              +              1                                                                        =                              1            n                                                              n              +              2                        3                                                            =                  1          3                                      n            +            2                    n                    Taking it to the limit, we get      ∏          j      =      2        ∞        (    1    −          2              j        (        j        +        1        )              )    =      1    3

django0a6 · Accepted Answer

You start with             1      ⋅      4              2      ⋅      3                  1      ⋅      4              2      ⋅      3        ×            2      ⋅      5              3      ⋅      4        =            1      ⋅      5              3      ⋅      3      and adding another term            1      ⋅      5              3      ⋅      3        ×            3      ⋅      6              4      ⋅      5        =            1      ⋅      6              3      ⋅      4      and proceed by induction to prove that       x    n    =            1      ⋅      (      n      +      2      )              3      ⋅      n       which in the limit should converge to 1/3.

Find the limit of x_n=prod_(j=2)^n (1-2/(j(j+1)))^2

Answered question

Answer & Explanation

New Questions in Calculus 2