Write for convenience. Then , where the sum is taken over all odd values of (the positive integer) n. Multiply both sides by to see that if the sum is taken over all odd multiples of 2. Then if the sum is taken over all odd multples of 4, and so on. Since every positive integer is (uniquely) equal to some power of 2 times an odd integer, it follows that , where the sum is now taken over all positive integers. Sum the geometric series to get the result.