Given an infinite sequence , the nth partial sum is the sum of the first n terms of the sequence, that is,
A series is convergent if the sequence of its partial sums tends to a limit; that means that the partial sums become closer and closer to a given number when the number of their terms increases. More precisely, a series converges, if there exists a number such that for any arbitrarily small positive number , there is a (sufficiently large) integer such that for all ,
If the series is convergent, the number (necessarily unique) is called the sum of the series.
Any series that is not convergent is said to be divergent.
There are a number of methods of determining whether a series converges or diverges.
If the blue series, , can be proven to converge, then the smaller series, must converge. By contraposition, if the red series, is proven to diverge, then must also diverge.
Comparison test. The terms of the sequence are compared to those of another sequence . If,
for all n, , and converges, then so does
for all n, , and diverges, then so does
Ratio test. Assume that for all n, . Suppose that there exists such that
If r < 1, then the series converges. If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge.
Root test or nth root test. Suppose that the terms of the sequence in question are non-negative. Define r as follows:
where "lim sup" denotes the limit superior (possibly ∞; if the limit exists it is the same value).
If r < 1, then the series converges. If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge.
The ratio test and the root test are both based on comparison with a geometric series, and as such they work in similar situations. In fact, if the ratio test works (meaning that the limit exists and is not equal to 1) then so does the root test; the converse, however, is not true. The root test is therefore more generally applicable, but as a practical matter the limit is often difficult to compute for commonly seen types of series.
Illustration of the absolute convergence of the power series of Exp[z] around 0 evaluated at z = Exp[i⁄3]. The length of the line is finite.
Illustration of the conditional convergence of the power series of log(z+1) around 0 evaluated at z = exp((π−1⁄3)i). The length of the line is infinite.
For any sequence , for all n. Therefore,
This means that if converges, then also converges (but not vice versa).
If the series converges, then the series is absolutely convergent. An absolutely convergent sequence is one in which the length of the line created by joining together all of the increments to the partial sum is finitely long. The power series of the exponential function is absolutely convergent everywhere.
If the series converges but the series diverges, then the series is conditionally convergent. The path formed by connecting the partial sums of a conditionally convergent series is infinitely long. The power series of the logarithm is conditionally convergent.
The Riemann series theorem states that if a series converges conditionally, it is possible to rearrange the terms of the series in such a way that the series converges to any value, or even diverges.