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Trigonometry/The summation of finite series

< Trigonometry

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




Similarly, if




Method 2Edit

Consider the following sum


Since   is a geometric series with common ratio   , we get