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

.

Hence

.

Similarly, if

then

.

Method 2[edit]

Consider the following sum

.

Since is a geometric series with common ratio , we get

Therefore,