arctan ⁡ ( 1 3 ) + arctan ⁡ ( 1 7 ) + arctan ⁡ ( 1 13 ) + arctan ⁡ ( 1 21 ) + ⋯Estimate the value of the expression if n → ∞, how to derive this and what will be the approach of this kind of question ?I have tried to solve this by first doing partial sums then taking limit but I can't evaluate this after getting the form ( n 2 + n + 1 )

Question

arctan  ⁡      (          1      3        )    +  arctan  ⁡      (          1      7        )    +  arctan  ⁡      (          1      13        )    +  arctan  ⁡      (          1      21        )    +  ⋯Estimate the value of the expression if   n  →  ∞, how to derive this and what will be the approach of this kind of question ?I have tried to solve this by first doing partial sums then taking limit but I can&#039;t evaluate this after getting the form       (          n              2              +    n    +    1    )

Josie123 · Accepted Answer

arctan  ⁡  (  x  )  −  arctan  ⁡  (  y  )  =  arctan  ⁡      (                  x        −        y                    1        +        x        y              )  Now use x=n+1 and y=n and you will get a telescoping sum.

Roland Waters · Accepted Answer

lim                      N            →            ∞                                                            ∑                      k            =            0                                N                          arctan        ⁡                  (                                              1                                                k                                      2                                                  +                3                k                +                3                                                        )                =                  lim                      N            →            ∞                                                            ∑                      k            =            0                                N                                    [                                arctan            ⁡                          (                                            k                +                2                                          )                        −            arctan            ⁡                          (                                            k                +                1                                          )                                          ]                                    =                                       lim                      N            →            ∞                                    {                                                                      A                                                                          [                                                                                                                  A                                                                                  arctan              ⁡                              (                                                  2                                                )                            −              arctan              ⁡                              (                                                  1                                                )                                                  ]                    +                      [                                                                                                                  A                                                                                  arctan              ⁡                              (                                                  3                                                )                            −              arctan              ⁡                              (                                                  2                                                )                                                  ]                    +          ⋯                                                                                                                           lim                                  N                  →                  ∞                                                                        +                      [                                                                                                                  A                                                                                  arctan              ⁡                              (                                                  N                  +                  2                                                )                            −              arctan              ⁡                              (                                                  N                  +                  1                                                )                                                  ]                    }                                    =                                       lim                      N            →            ∞                                                            [                                                                                                    A                                                                        −            arctan            ⁡                          (                                            1                                          )                        +            arctan            ⁡                          (                                            N                +                2                                          )                                          ]                =        −                                      π            4                          +                              π            2                          =                                                                                                          π                    4

arctan ⁡ ( 1 3 </mfrac> ) </mrow>

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