Trigonometry/The summation of finite series

Problem Statement 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 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 2 Consider the following sum Since is a geometric series with common ratio , we get Therefore,