Trigonometry/The summation of finite series

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Problem StatementEdit

Find a closed form for
  .

Note: A 'closed form' is not mathematically defined, but just means a simplified formula which does not involve '...', or a summation sign. In our problem, we should look for a formula that only involves variables   , and known operations like the four operations, radicals, exponents, logarithm, and trigonometric functions.

Method 1Edit

To sum the series

  .

Multiply each term by

  .

Then we have

 

and similarly for all terms to

  .

Summing, we find that nearly all the terms cancel out and we are left with

  .

Hence

  .

Similarly, if

 

then

  .

Method 2Edit

Consider the following sum

  .

Since   is a geometric series with common ratio   , we get

 
 

Therefore,