Evaluate lim n → ∞ sin 2 ⁡ ( π n 2 + n ) , n ∈ N

Question

Evaluate       lim          n      →      ∞            sin    2    ⁡  (  π            n      2        +    n    )  ,  n  ∈      N

Samuel Vang · Accepted Answer

First of all, notice that the function   f  (  θ  )  =      sin    2    ⁡  θ is periodic with a period π. Then, for any integer n we should have       sin    2    ⁡  (  π            n      2        +    n    )  =      sin    2    ⁡      (    π          (                        n          2                +        n            −      n      )        )  . Since      lim          n      →      ∞            (                  n        2            +      n        −    n    )    =      1    2    ,it becomes obvious that      lim          n      →      ∞            sin    2    ⁡  (  π            n      2        +    n    )  =      sin    2    ⁡      π    2    =  1.

Aditya Erickson · Accepted Answer

n      2        +    n    =  n      1    +          1      n        =  n  (  1  +      1          2      n        +  O  (  1      /        n    2    )  )  =  =  n  +      1    2    +  O  (  1      /    n  )

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