arccot limit: ∑ r = 1 ∞ cot − 1 ⁡ ( r 2 + 3 4 )

Question

arccot limit:       ∑          r      =      1              ∞            cot          −      1        ⁡  (      r    2    +      3    4    )

talhekh · Accepted Answer

HintThe general term can be written as      tan          −      1        ⁡      1                  r        2            +      3              /            4        =      tan          −      1        ⁡            r      +      1              /            2      −      (      r      −      1              /            2      )              (      r      −      1              /            2      )      (      r      +      1              /            2      )      +      1        =      tan          −      1        ⁡  (  r  +  1      /    2  )  −      tan          −      1        ⁡  (  r  −  1      /    2  )

nidantasnu · Accepted Answer

One may use  arctan  ⁡  a  −  arctan  ⁡  b  =  arctan  ⁡      (                  a        −        b                    1        +        a        b              )    ,    a  ,  b  ∈      [    0    ,          π      2        ]    ,with  a  =            2      n      −      3              4      (      n      +      1      )        ,    b  =            2      (      n      −      1      )      −      3              4      n        ,    n  =  1  ,  2  ,  3  ,  …  ,giving, for   n  ≥  1,  arctan  ⁡      (                  2        n        −        3                    4        (        n        +        1        )              )    −  arctan  ⁡      (                  2        (        n        −        1        )        −        3                    4        n              )    =  arctan  ⁡      (          1                        n          2                +                  3          4                      )  then, by telescoping,      ∑          n      =      1        N    arctan  ⁡      (          1                        n          2                +                  3          4                      )    =  arctan  ⁡      (                  2        N        −        3                    4        (        N        +        1        )              )    −  arctan  ⁡      (    −          3      4        )    .Letting   N  →  ∞ gives      ∑          n      =      1        ∞    arctan  ⁡      (          1                        n          2                +                  3          4                      )    =  arctan  ⁡      (          1      2        )    +  arctan  ⁡      (          3      4        )    =  arctan  ⁡  2.

arccot limit: <munderover> ∑ <mrow class="MJX-TeXAtom-ORD"> r

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