Find the value of lim n → ∞ ∑ r = 2 n ( cot ⁡ ( S r − 1 ) − cot ⁡ ( S r ) ) where S n = ∑ r = 1 n tan − 1 ⁡ ( 2 ( 2 r − 1 ) 4 + r 2 ( r 2 − 2 r + 1 ) )

Question

Find the value of      lim          n      →      ∞            ∑          r      =      2        n        (    cot    ⁡          (              S                  r          −          1                    )        −    cot    ⁡          (              S        r            )        )  where      S    n    =      ∑          r      =      1        n        tan          −      1        ⁡      (                  2                  (          2          r          −          1          )                            4        +                  r          2                          (                      r            2                    −          2          r          +          1          )                      )

ejigaboo8y · Accepted Answer

Let       a    r    =      tan          −      1        ⁡      (                            r          2                2              )              a      r        −          a              r        −        1              =          tan              −        1              ⁡          (                        2                      (            2            r            −            1            )                                    4          +                      r            2                                (                          r              2                        −            2            r            +            1            )                              )      Therefore,       S    n  , like the limit you are trying to find, are just a telescoping series.

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