Proving the Collatz Conjecture

Collatz sequences are formed by dividing an even number by two until it is odd. Then multiply by three and add one to get an even number. The Collatz conjecture states that if this process is repeated you always get back to one. Using geometric series summations we prove that a connected Collatz Structure exists, which contains all positive integers exactly once. The terms of the Collatz Structure are joined together via the Collatz algorithm. Thus, every positive integer forms a Collatz sequence with unique terms terminating in the number one.