How to simplify lim n → ∞ ∑ r = 1 n tan − 1 ⁡ 2 r + 1 r 4 + 2 r 3 + r 2 + 1

Question

How to simplify       lim          n      →      ∞            ∑          r      =      1        n        tan          −      1        ⁡                    2        r        +        1                              r          4                +        2                  r          3                +                  r          2                +        1

Bornejecbo · Accepted Answer

HintSince                    2        r        +        1                    (                  r          2                +        r                  )          2                +        1              =                    (        r        +        1                  )          2                −                  r          2                            1        +                  r          2                (        r        +        1                  )          2                    then      tan          −      1        ⁡            (                          2        r        +        1                    (                  r          2                +        r                  )          2                +        1                        )        =      tan          −      1        ⁡  (  1  +  r      )    2    −      tan          −      1        ⁡  (      r    2    )which beautifully telecospes.

How to simplify <munder> <mo movablelimits="true" form="prefix">lim <mrow class="MJX-T

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