Show that lim n → ∞ ∏ i = n B n arctan ⁡ ( i ϕ ) arccos ⁡ ( ϕ i ) = B 2 π

Question

Show that       lim          n      →      ∞            ∏          i      =      n              B      n                  arctan      ⁡      (      i      ϕ      )              arccos      ⁡              (                  ϕ          i                )              =      B                  2        π

britspears523jp · Accepted Answer

Denote   f  (  x  )  =  log  ⁡            arctan      ⁡      (      x      ϕ      )              arccos      ⁡              (                  ϕ          x                )            . Expanding into series  f  (  x  )  =      2          π      x        +  O  (      x          −      2        )  ,    x  →  ∞  ,and using the Euler&#039;s approximation for the harmonic sums      ∑          i      =      1        n        1    i    =  γ  +  log  ⁡  n  +  O  (  1      /    n  )  ,where   γ is the Euler&#039;s constant, gives  log  ⁡      lim          n      →      ∞            ∏          i      =      n              B      n                  arctan      ⁡      (      i      ϕ      )              arccos      ⁡              (                  ϕ          i                )              =      lim          n      →      ∞            ∑          i      =      n              B      n        f  (  i  )  =      lim          n      →      ∞            [          2      π              (              ∑                  i          =          1                          B          n                            1        i            −              ∑                  i          =          1                          n          −          1                            1        i            )        +    O    (          n              −        1              )    ]    =  =      lim          n      →      ∞            [          2      π        (    log    ⁡    B    n    −    log    ⁡    n    )    +    O    (          n              −        1              )    ]    =      2    π    log  ⁡  B  .

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