## 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,