Trigonometry/The summation of finite series

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Problem Statement[edit]

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 1[edit]

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




Similarly, if



Method 2[edit]

Consider the following sum


Since is a geometric series with common ratio , we get