Let n be a non-negative integer and let f ( x ) = ∑ k = 0 n ( n k ) sin k ⁡ x sec k − n ⁡ x .Prove that f(x) is periodic and find its amplitude.

Question

Let n be a non-negative integer and let  f  (  x  )  =      ∑          k      =      0              n                                    (                            n        k                              )                          sin          k        ⁡  x      sec          k      −      n        ⁡      x    .Prove that f(x) is periodic and find its amplitude.

Audrey Russell · Accepted Answer

Hint: Use the binomial theorem:  (  x  +  y      )    n    =      ∑          k      =      0        n                      (                    n      k                      )                  x    k        y          n      −      k      You can express the proposed sum as                    f        (        x        )                            =                  ∑                      k            =            0                                n                                                                              (                                                    n              k                                                      )                                                              sin                      k                          ⁡        (        x        )                  sec                      k            −            n                          ⁡        (        x        )                                          =                  ∑                      k            =            0                                n                                                                              (                                                    n              k                                                      )                                                              sin                      k                          ⁡        (        x        )                  cos                      n            −            k                          ⁡        (        x        )                                                        =        (        sin        ⁡        (        x        )        +        cos        ⁡        (        x        )                  )                      n                                                                          =                  2                      n                          /                        2                                                (                          1                              2                                      sin            ⁡            (            x            )            +                          1                              2                                      cos            ⁡            (            x            )            )                                n                                                                          =                  2                      n                          /                        2                                    sin                      n                          ⁡                  (          x          +                      π            4                    )

Let n be a non-negative integer and let f(x)=sum_(k=0)^n ((n),(k)) sin^k x sec^(k−n) x. Prove that f(x) is periodic and find its amplitude

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