Trigonometric Summation ∑ r = 0 ∞ sin ⁡ ( r θ ) 3 r

Question

Trigonometric Summation       ∑          r      =      0              ∞                  sin      ⁡      (      r              θ            )              3      r

gutinyalk · Accepted Answer

The sum is  ℑ      ∑          r      =      0        ∞              (              1        3            exp      ⁡      i      θ      )        r    =  ℑ      1          1      −              1        3            exp      ⁡      i      θ        =  ℑ            1      −              1        3            exp      −      i      θ                      10        9            −              2        3            cos      ⁡      θ        =            3      sin      ⁡      θ              10      −      6      cos      ⁡      θ        .For   sin  ⁡  θ  =      1    3  ,   cos  ⁡  θ  =  ±            2              2              3  , so the sum simplifies to       1          10      ∓      4              2              =            5      ±      2              2              34  .

Riya Hansen · Accepted Answer

Use geometric series. We have                              ∑                      r            =            0                    ∞                                      sin            ⁡            (            r            θ            )                                3            r                                              =                  ∑                      r            =            0                    ∞                          I          m                          (                                    e                              i                r                θ                                                    3              r                                )                                                  =                  I          m                          (                      ∑                          r              =              0                        ∞                                              (                                                e                                      i                    θ                                                  3                            )                        r                    )                                                  =                  I          m                          (                      1                          1              −                              e                                  i                  θ                                                            /                            3                                )                                                  =                  I          m                          (                                    1              +                              e                                  −                  i                  θ                                                            /                            3                                      1              +              (                              e                                  i                  θ                                            −                              e                                  −                  i                  θ                                            )                              /                            3              −              1                              /                            9                                )                                                  =                  I          m                          (                                    1              +              cos              ⁡              (              θ              )                              /                            3              −              i              sin              ⁡              (              θ              )                              /                            3                                      8                              /                            9              +              2              sin              ⁡              (              θ              )                              /                            3                                )                                                  =                              −            1                          /                        9                                8                          /                        9            +            2                          /                        9                                                            =        −                  1          10                    if I didn&#039;t make a silly mistake somewhere.

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