In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, an alternating series is an infinite series of the form


with a_n ≥ 0 (or a_n ≤ 0) for all n. Like any series, an alternating series converges if and only if the associated sequence of partial sums converges

Limit of a sequence

The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...

.

Alternating series test

The theorem known as "Leibniz Test" or the alternating series test

Alternating series test

The alternating series test is a method used to prove that infinite series of terms converge. It was discovered by Gottfried Leibniz and is sometimes known as Leibniz's test or the Leibniz criterion.A series of the form...

 tells us that an alternating series will converge if the terms a_n converge to 0 monotonically

Monotonic function

In mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....

.

Proof: Suppose the sequence converges to zero and is monotone decreasing. If is odd and , we obtain the estimate via the following calculation:

Since is monotonically decreasing, the terms are negative. Thus, we have the final inequality . Since converges to , our partial sums form a cauchy sequence

Cauchy sequence

In mathematics, a Cauchy sequence , named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses...

 (i.e. the series satisfies the cauchy convergence criterion for series) and therefore converge. The argument for even is similar.

Approximating sums

The estimate above does not depend on . So, if is approaching 0 monotonically, the estimate provides an error bound for approximating infinite sums by partial sums:

Absolute convergence

A series converges absolutely

Absolute convergence

In mathematics, a series of numbers is said to converge absolutely if the sum of the absolute value of the summand or integrand is finite...

 if the series converges.

Theorem: Absolutely convergent series are convergent.

Proof: Suppose is absolutely convergent. Then, is convergent and it follows that converges as well. Since , the series converges by the comparison test

Comparison test

In mathematics, the comparison test, sometimes called the direct comparison test or CQT is a criterion for convergence or divergence of a series whose terms are real or complex numbers...

. Therefore, the series converges as the difference of two convergent series .

Conditional convergence

A series is conditionally convergent if it converges but does not converge absolutely.

For example, the harmonic series

Harmonic series (mathematics)

In mathematics, the harmonic series is the divergent infinite series:Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 1/4, etc., of the string's fundamental wavelength...

diverges, while the alternating version


converges by the alternating series test.

Rearrangements

For any series, we can create a new series by rearranging the order of summation. A series is unconditionally convergent if any rearrangement creates a series with the same convergence as the original series. Absolutely convergent series are unconditionally convergent. But the Riemann series theorem states that conditionally convergent series can be rearranged to create arbitrary convergence. The general principle is that addition of infinite sums is only associative for absolutely convergent series.

For example, this false proof that 1=0 exploits the failure of associativity for infinite sums.

As another example, we know that .

But, since the series does not converge absolutely, we can rearrange the terms to obtain a series for :

Series acceleration

In practice, the numerical summation of an alternating series may be sped up using any one of a variety of series acceleration

Series acceleration

In mathematics, series acceleration is one of a collection of sequence transformations for improving the rate of convergence of a series. Techniques for series acceleration are often applied in numerical analysis, where they are used to improve the speed of numerical integration...

 techniques. One of the oldest techniques is that of Euler summation

Euler summation

Euler summation is a summability method for convergent and divergent series. Given a series Σan, if its Euler transform converges to a sum, then that sum is called the Euler sum of the original series....

, and there are many modern techniques that can offer even more rapid convergence.

The source of this article is wikipedia, the free encyclopedia.  The text of this article is licensed under the GFDL.