In addition to ordered fields which are not complete, there are complete fields which are not ordered. The complex numbers are perhaps the best known example.
But there are also complete "nonArchimedian" fields. Their topology is defined by a "valuation" which takes nonnegative real values, and satisfies
x*y = x*y, and x + y <= Max(x, y).
The basic examples are the padic completions of the rational numbers, where p is a prime number.
The padic valuation of a nonzero integer n is p^{e}, where p^{e} is the exact power of p dividing n. The padic valuation of 0 is 0. The padic valuation of any nonmultiple of p is 1, and integers divisible by high powers of p are "small."
Thus, in the 2adic rationals (and its completion), it is perfectly correct to say that the "infinite sum" of powers of 2
1 + 2 + 4 + 8 + ... = 1
