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Expand and evaluate each of the following sums.

  1. ___

    or, because of the symmetry of , this problem can be simplified as:

  1. Think about the symmetry of odd functions such as .
    Justify why the sum from  to will be .

  1. Unfortunately, does not have symmetry across the -axis. We have to expand and evaluate all terms.

  1. sin is an odd function. Consider symmetry before evaluating.