Working out the properties of the series that converge even if infinitely many terms are non-zero is the essence of the study of series. Consider the example  It is possible to "visualize" its convergence on the real number line: we can imagine a line of length 2, with successive segments marked off of lengths 1, ½, ¼, etc. There is always room to mark the next segment, because the amount of line remaining is always the same as the last segment marked: when we have marked off ½, we still have a piece of length ½ unmarked, so we can certainly mark the next ¼. This argument does not prove that the sum is "equal" to 2 (although it is), but it does prove that it is "at most" 2. In other words, the series has an upper bound. Given that the series converges, proving that it is equal to 2 requires only elementary algebra. If the series is denoted , it can be seen that  Therefore,  The idiom can be extended to other, equivalent notions of series. For instance, a recurring decimal, as in  encodes the series  Since these series always converge to real numbers (because of what is called the completeness property of the real numbers), to talk about the series in this way is the same as to talk about the numbers for which they stand. In particular, the decimal expansion 0.111... can be identified with 1/9. This leads to an argument that , which only relies on the fact that the limit laws for series preserve the arithmetic operations; this argument is presented in the article 0.999...  and their generalizations (such as basic hypergeometric series and elliptic hypergeometric series) frequently appear in integrable systems and mathematical physics.  and   Partial summation takes as input a sequence, ("a""n"), and gives as output another sequence, ("S""N"). It is thus a unary operation on sequences. Further, this function is linear, and thus is a linear operator on the vector space of sequences, denoted Σ. The inverse operator is the finite difference operator, denoted Δ. These behave as discrete analogs of integration and differentiation, only for series (functions of a natural number) instead of functions of a real variable. For example, the sequence (1, 1, 1, ...) has series (1, 2, 3, 4, ...) as its partial summation, which is analogous to the fact that formula_49  In computer science it is known as prefix sum.