Series

A series is a sum of the terms of a sequence. An infinite series has infinitely many terms. The order of the terms in a series can matter, since the Riemann series theorem states that, by a suitable rearrangement of terms, a so-called conditionally convergent series of real numbers may be made to converge to any desired real value, or to diverge.

Conditions for convergence of a series can be determined in the Wolfram Language using SumConvergence[a, n].

If the difference between successive terms of a series is a constant, then the series is said to be an arithmetic series. A series for which the ratio of each two consecutive terms is a constant function of the summation index is called a geometric series. The more general case of the ratio being a rational function of produces a series called a hypergeometric series.

A series may converge to a definite value, or may not, in which case it is called divergent. Let the terms in a series be denoted , let the th partial sum be given by

(1)

and let the sequence of partial sums be given by . If the sequence of partial sums converges to a definite value, the series is said to converge. On the other hand, if the sequence of partial sums does not converge to a limit (e.g., it oscillates or approaches ), the series is said to diverge. An example of a convergent series is the geometric series

(2)

and an example of a divergent series is the harmonic series

(3)

Interestingly, while the harmonic series diverges to infinity, the alternating harmonic series converges to the natural logarithm of 2,

(4)

Another well-known convergent series defines Brun's constant.

A number of methods known as convergence tests can be used to determine whether a given series converges. Although terms of a series can have either sign, convergence properties can often be computed in the "worst case" of all terms being positive, and then applied to the particular series at hand. A series of terms is said to be absolutely convergent if the series formed by taking the absolute values of the ,

(5)

converges.

An especially strong type of convergence is called uniform convergence, and series which are uniformly convergent have particularly "nice" properties. For example, the sum of a uniformly convergent series of continuous functions is continuous. A convergent series can be differentiated term by term on an interval, provided that the functions have continuous derivatives and the series of derivatives is uniformly convergent. Finally, on a bounded closed interval, a uniformly convergent series of continuous functions can be integrated term by term.

For a table listing the coefficients for various series operations, see Abramowitz and Stegun (1972, p. 15).

While it can be difficult to calculate analytical expressions for arbitrary convergent infinite series, many algorithms can handle a variety of common series types. The Wolfram Language computational system implements many of these algorithms. General techniques also exist for computing the numerical values of any but the most pathological series (Braden 1992).

A particular infinite series identity is given by

			(6)
			(7)

for and real .

Apostol (1997, p. 25) gives the analytic sum

(8)

for a nonnegative integer , where is a Bernoulli number.

Ramanujan found the interesting series identity

(9)

(Preece 1928; Hardy 1999, p. 7), which can be written as the hypergeometric identity

(10)

Positive integer powers of a geometric series can also be computed analytically,

			(11)
			(12)
			(13)

for a positive integer and , where is a Pochhammer symbol.

Gosper noted the sum

			(14)
			(15)
			(16)

(OEIS A100074).

An infinite series of the following form can be done in closed form.

(17)

for a positive integer , where is a polynomial. The first few values are

			(18)
			(19)
			(20)
			(21)

(OEIS A085470).

The related infinite series

(22)

for a positive integer can also be done in closed form, where is a polynomial. The first few values are

			(23)
			(24)
			(25)
			(26)
			(27)

(OEIS A085471).