An infinite series or simply a series is an infinite sum, represented by an infinite expression of the form
where is any ordered sequence of terms, such as numbers, functions, or anything else that can be added (an abelian group). This is an expression that is obtained from the list of terms by laying them side by side, and conjoining them with the symbol "+". A series may also be represented by using summation notation, such as
If an abelian group A of terms has a concept of limit (for example, if it is a metric space), then some series, the convergent series, can be interpreted as having a value in A, called the sum of the series. This includes the common cases from calculus in which the group is the field of real numbers or the field of complex numbers. Given a series its kth partial sum is
By definition, the series converges to the limit L (or simply sums to L), if the sequence of its partial sums has a limit L. In this case, one usually writes
A series is said to be convergent if it converges to some limit or divergent when it does not. The value of this limit, if it exists, is then the value of the series.
Illustration of 3 geometric series
with partial sums from 1 to 6 terms. The dashed line represents the limit.
A series ∑an is said to converge or to be convergent when the sequence (sk) of partial sums has a finite limit. If the limit of sk is infinite or does not exist, the series is said to diverge. When the limit of partial sums exists, it is called the value (or sum) of the series
An easy way that an infinite series can converge is if all the an are zero for n sufficiently large. Such a series can be identified with a finite sum, so it is only infinite in a trivial sense.
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 S, it can be seen that
Mathematicians extend the idiom discussed earlier to other, equivalent notions of series. For instance, when we talk about a recurring decimal, as in
we are talking, in fact, just about the series
But 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 9 × 0.111… = 0.999… = 1, 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....