Prove that ∑ j = − N N e i ⋅ j ⋅ ξ ⋅ λ = sin ⁡ ( N + 1 2 ) ⋅ ( ξ ⋅ λ ) sin ⁡ ( ξ ⋅ λ 2 )

Question

Prove that      ∑          j      =      −      N        N        e          i      ⋅      j      ⋅              ξ            ⋅              λ              =                    sin        ⁡        (        N        +                              1            2                          )        ⋅        (                  ξ                ⋅                  λ                )                    sin        ⁡        (                                                            ξ                            ⋅                              λ                                      2                          )

rass1k6s · Accepted Answer

To simplify, let   ξ  ⋅  λ  =  x                              ∑                      j            =            −            N                    N                          e          i                j                    x        =                  e                      −            i            N            x                                                              1              −                              e                                  2                  i                  N                  x                                                                    1              −                              e                                  i                  x                                                                    =                  e                      −            i            N            x                                                              e                              i                (                N                +                1                                  /                                2                )                x                                                    e                              i                x                                  /                                2                                                                                                        e                                  −                  i                  (                  N                  +                  1                                      /                                    2                  )                  x                                            −                              e                                  i                  (                  N                  +                  1                                      /                                    2                  )                  x                                                                                    e                                  −                  i                  x                                      /                                    2                                            −                              e                                  i                  x                                      /                                    2                                                                                                      =                                            sin              ⁡              (              (              N              +              1                              /                            2              )              x              )                                      s              i              n              (              x                              /                            2              )                                      .

Prove that <munderover> ∑ <mrow class="MJX-TeXAtom-ORD"> j

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Answer & Explanation

New Questions in Calculus 2