I need to find the limit for x ∈ R lim n → ∞ ∑ k = 0 n | e i x k / n − e i x ( k − 1 ) / n | Using Euler's formula and cosine of a sum this simplifies to lim n → ∞ n 2 ( 1 − cos ⁡ ( x / n ) )

Question

I need to find the limit for   x  ∈      R        lim          n      →      ∞            ∑          k      =      0        n        |        e          i      x      k              /            n        −      e          i      x      (      k      −      1      )              /            n            |  Using Euler&#039;s formula and cosine of a sum this simplifies to      lim          n      →      ∞        n      2    (    1    −    cos    ⁡    (    x          /        n    )    )

Megan Mathis · Accepted Answer

One has the Taylor expansion   cos  ⁡  x  =  1  −      1    2        x    2    +  O  (      x    3    ), so in particular for fixed x we have  2  (  1  −  cos  ⁡  (  x      /    n  )  )  =            x      2              n      2        +  O  (      n          −      3        )  .

Hailey Newton · Accepted Answer

Notice that fix x  cos  ⁡  (  2  x  )  =  1  −  2      sin    2    ⁡  (  x  )      lim          n      →      ∞        n      2    (    1    −    cos    ⁡    (    x          /        n    )    )    =      lim          n      →      ∞        n      4          sin      2        ⁡    (          x              2        n              )    =      lim          n      →      ∞        2  n      |    sin  ⁡  (      x          2      n        )      |    =      lim          n      →      ∞        2  n                    |            x              |                    2      n        =      |    x      |  EDITFor the second last equality:  sin  ⁡  (  x  )  =  x  +  O  (      x    2    ) (By Tayler expansion around 0,   sin  ⁡  x  =  x  −      1    6        x    3    +  .  .  .)

I need to find the limit for x ∈ <mrow class="MJX-TeXAtom-ORD"> <mi

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