Evaluate ∑ n = 1 ∞ sin ⁡ ( a 3 n ) sin ⁡ ( 2 a 3 n ) where a is just some real constant.

Question

Evaluate        ∑          n      =      1              ∞        sin  ⁡            (              a                  3          n                    )        sin  ⁡            (                        2          a                          3          n                    )      where a is just some real constant.

Ronald Hickman · Accepted Answer

HINT: Notice      ∑          n      =      1              ∞        sin  ⁡      (          a              3        n              )    sin  ⁡      (                  2        a                    3        n              )    =      ∑          n      =      1              ∞            1    2        (    2    sin    ⁡          (              a                  3          n                    )        sin    ⁡          (                        2          a                          3          n                    )        )    =      ∑          n      =      1              ∞            1    2        (    cos    ⁡          (              a                  3          n                    −                        2          a                          3          n                    )        −    cos    ⁡          (              a                  3          n                    +                        2          a                          3          n                    )        )    =      1    2        ∑          n      =      1              ∞            (    cos    ⁡          (              a                  3          n                    )        −    cos    ⁡          (              a                  3                      n            −            1                              )        )

Gretchen Schwartz · Accepted Answer

Use   sin  ⁡  a  sin  ⁡  b  =      1    2    (  cos  ⁡  (  a  −  b  )  −  cos  ⁡  (  a  +  b  )  ). Expand the summation, it will be a telescoping series. All terms except 2 will cancel out. As n becomes large, one of the terms will tend to 1, the other one will be   cos  ⁡  a. The final answer should be   1  −  cos  ⁡  a

Evaluate <mspace width="1em" /> <munderover> ∑ <mrow class="MJX-

Answered question

Answer & Explanation

New Questions in Precalculus