I am considering the sum A m = ∑ j = 0 m sin 4 ⁡ ( j m ⋅ π 2 ) .I think that for m&gt;1 it holds A m = 3 m + 4 8 ,but I can't really get to it.

Question

I am considering the sum      A    m    =      ∑          j      =      0        m        sin    4    ⁡      (          j      m        ⋅          π      2        )    .I think that for m&amp;gt;1 it holds      A    m    =            3      m      +      4        8    ,but I can&#039;t really get to it.

Lea Johnson · Accepted Answer

Start by linearizing       sin          4        ⁡  (  x  ) :  ∀  x  ∈      R    ,        sin          4        ⁡  (  x  )  =      1    8              (        cos  ⁡  (  4  x  )  −  4  cos  ⁡  (  2  x  )  +  3            )        .This leads to :      ∑          j      =      1              m            sin          4        ⁡            (                  j      π              2      m                  )        =      1    8              (            ∑          j      =      1              m        cos  ⁡            (                  2      j      π        m              )        −  4      ∑          j      =      1              m        cos  ⁡            (                  j      π        m              )        +  3  m            )        .The sums             ∑              j        =        1                    m              cos    ⁡                  (                            2        j        π            m                      )             and             ∑              j        =        1                    m              cos    ⁡                  (                            j        π            m                      )             can be computed using complex numbers. For   z  ∈      C  ,   z  ∈      C   denotes the real part of z. Indeed, note that :      ∑          j      =      1              m        cos  ⁡            (                  2      j      π        m              )        =  ℜ            (            ∑          j      =      1              m            e          2      i      j      π              /            m                  )        .and, since   exp  ⁡  (  2  i  π      /    m  )  ≠  1 for all m, we have :      ∑          j      =      1              m            e          2      i      j      π              /            m        =      e          2      i      π              /            m                  1      −                        (                            e                  2          i          π                      /                    m                                                  )                                    m                            1      −              e                  2          i          π                      /                    m                      =  0.Similarily :             ∑              j        =        1                    m              cos    ⁡                  (                            j        π            m                      )              =    −    1  As a consequence :      ∑          j      =      1              m            sin          4        ⁡            (                  j      π              2      m                  )        =      1    8    (  3  m  +  4  )  .

I am considering the sum A m </msub> = <munderover> ∑<!-- ∑ --

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