Alternating series

Short description: Infinite series whose terms alternate in sign

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In mathematics, an alternating series is an infinite series of terms that alternate between positive and negative signs. In capital-sigma notation this is expressed ∑n=0∞(−1)nan or ∑n=0∞(−1)n+1an with an > 0 for all n.

Like any series, an alternating series is a convergent series if and only if the sequence of partial sums of the series converges to a limit. The alternating series test guarantees that an alternating series is convergent if the terms an converge to 0 monotonically, but this condition is not necessary for convergence.

## Contents

* 1 Examples
* 2 Alternating series test
* 3 Approximating sums
* 4 Absolute convergence
* 5 Conditional convergence
* 6 Rearrangements
* 7 Series acceleration
* 8 See also
* 9 Notes
* 10 References

## Examples

The geometric series ⁠1/2⁠ − ⁠1/4⁠ \+ ⁠1/8⁠ − ⁠1/16⁠ \+ ⋯ sums to ⁠1/3⁠.

The alternating harmonic series has a finite sum but the harmonic series does not. The series 1−13+15−…=∑n=0∞(−1)n2n+1 converges to π4, but is not absolutely convergent.

The Mercator series provides an analytic power series expression of the natural logarithm, given by ∑n=1∞(−1)n+1nxn=ln⁡(1+x),|x|≤1,x≠−1.

The functions sine and cosine used in trigonometry and introduced in elementary algebra as the ratio of sides of a right triangle can also be defined as alternating series in calculus. sin⁡x=∑n=0∞(−1)nx2n+1(2n+1)! and cos⁡x=∑n=0∞(−1)nx2n(2n)!. When the alternating factor (–1)n is removed from these series one obtains the hyperbolic functions sinh and cosh used in calculus and statistics.

For integer or positive index α the Bessel function of the first kind may be defined with the alternating series Jα(x)=∑m=0∞(−1)mm!Γ(m+α+1)(x2)2m+α where Γ(z) is the gamma function.

If s is a complex number, the Dirichlet eta function is formed as an alternating series η(s)=∑n=1∞(−1)n−1ns=11s−12s+13s−14s+⋯ that is used in analytic number theory.

## Alternating series test

Main article: Alternating series test

The theorem known as the "Leibniz Test" or the alternating series test states that an alternating series will converge if the terms an converge to 0 monotonically.

Proof: Suppose the sequence an converges to zero and is monotone decreasing. If m is odd and m<n, we obtain the estimate Sn−Sm≤am via the following calculation: Sn−Sm=∑k=0n(−1)kak−∑k=0m(−1)kak =∑k=m+1n(−1)kak=am+1−am+2+am+3−am+4+⋯+an=am+1−(am+2−am+3)−(am+4−am+5)−⋯−an≤am+1≤am.

Since an is monotonically decreasing, the terms −(am−am+1) are negative. Thus, we have the final inequality: Sn−Sm≤am. Similarly, it can be shown that −am≤Sn−Sm. Since am converges to 0, the partial sums Sm form a Cauchy sequence (i.e., the series satisfies the Cauchy criterion) and therefore they converge. The argument for m even is similar.

## Approximating sums

The estimate above does not depend on n. So, if an is approaching 0 monotonically, the estimate provides an error bound for approximating infinite sums by partial sums: |∑k=0∞(−1)kak−∑k=0m(−1)kak|≤|am+1|.That does not mean that this estimate always finds the very first element after which error is less than the modulus of the next term in the series. Indeed if you take 1−1/2+1/3−1/4+...=ln⁡2 and try to find the term after which error is at most 0.00005, the inequality above shows that the partial sum up through a20000 is enough, but in fact this is twice as many terms as needed. Indeed, the error after summing first 9999 elements is 0.0000500025, and so taking the partial sum up through a10000 is sufficient. This series happens to have the property that constructing a new series with an−an+1 also gives an alternating series where the Leibniz test applies and thus makes this simple error bound not optimal. This was improved by the Calabrese bound,[1] discovered in 1962, that says that this property allows for a result 2 times less than with the Leibniz error bound. In fact this is also not optimal for series where this property applies 2 or more times, which is described by Johnsonbaugh error bound.[2] If one can apply the property an infinite number of times, Euler's transform applies.[3]

## Absolute convergence

A series ∑an converges absolutely if the series ∑|an| converges.

Theorem: Absolutely convergent series are convergent.

Proof: Suppose ∑an is absolutely convergent. Then, ∑|an| is convergent and it follows that ∑2|an| converges as well. Since 0≤an+|an|≤2|an|, the series ∑(an+|an|) converges by the comparison test. Therefore, the series ∑an converges as the difference of two convergent series ∑an=∑(an+|an|)−∑|an|.

## Conditional convergence

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

For example, the harmonic series

∑n=1∞1n

diverges, while the alternating version

∑n=1∞(−1)n+1n

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.[4] Agnew's theorem describes rearrangements that preserve convergence for all convergent series. The general principle is that addition of infinite sums is only commutative for absolutely convergent series.

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

As another example, by Mercator series ln⁡(2)=∑n=1∞(−1)n+1n=1−12+13−14+⋯.

But, since the series does not converge absolutely, we can rearrange the terms to obtain a series for 12ln⁡(2): (1−12)−14+(13−16)−18+(15−110)−112+⋯=12−14+16−18+110−112+⋯=12(1−12+13−14+15−16+⋯)=12ln⁡(2).

## Series acceleration

In practice, the numerical summation of an alternating series may be sped up using any one of a variety of series acceleration techniques. One of the oldest techniques is that of Euler summation, and there are many modern techniques that can offer even more rapid convergence.

## See also

* Grandi's series
* Nörlund–Rice integral

## Notes

1. ↑ Calabrese, Philip (March 1962). "A Note on Alternating Series". The American Mathematical Monthly 69 (3): 215–217. doi:10.2307/2311056. https://www.jstor.org/stable/2311056.
2. ↑ Johnsonbaugh, Richard (October 1979). "Summing an Alternating Series". The American Mathematical Monthly 86 (8): 637–648. doi:10.2307/2321292. https://www.jstor.org/stable/2321292.
3. ↑ Villarino, Mark B. (2015-11-27). "The error in an alternating series". arXiv:1511.08568 [math.CA].
4. ↑ Mallik, AK (2007). "Curious Consequences of Simple Sequences". Resonance 12 (1): 23–37. doi:10.1007/s12045-007-0004-7.

## References

* Earl D. Rainville (1967) Infinite Series, pp 73–6, Macmillan Publishers.
* Weisstein, Eric W.. "Alternating Series". http://mathworld.wolfram.com/AlternatingSeries.html.

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Sequences and series

Integer
sequences|

| Basic|

* Arithmetic progression
* Geometric progression
* Harmonic progression
* Square number
* Cubic number
* Factorial
* Powers of two
* Powers of 10

|

Advanced (list)|

* Complete sequence
* Fibonacci numbers
* Figurate number
* Heptagonal number
* Hexagonal number
* Lucas number
* Pell number
* Pentagonal number
* Polygonal number
* Triangular number

Properties of sequences|

* Cauchy sequence
* Monotone sequence
* Periodic sequence

Properties of series|

* Convergent series
* Divergent series
* Conditional convergence
* Absolute convergence
* Uniform convergence
* Alternating series
* Telescoping series

Explicit series|

| Convergent|

* 1/2 − 1/4 + 1/8 − 1/16 + ⋯
* 1/2 + 1/4 + 1/8 + 1/16 + ⋯
* 1/4 + 1/16 + 1/64 + 1/256 + ⋯
* 1 + 1/2s\+ 1/3s \+ ... (Riemann zeta function)

|

Divergent|

* 1 + 1 + 1 + 1 + ⋯
* 1 + 2 + 3 + 4 + ⋯
* 1 + 2 + 4 + 8 + ⋯
* 1 − 1 + 1 − 1 + ⋯ (Grandi's series)
* Infinite arithmetic series
* 1 − 2 + 3 − 4 + ⋯
* 1 − 2 + 4 − 8 + ⋯
* 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
* 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
* 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)

Kinds of series|

* Taylor series
* Power series
* Formal power series
* Laurent series
* Puiseux series
* Dirichlet series
* Trigonometric series
* Fourier series
* Generating series

Hypergeometric
series|

* Generalized hypergeometric series
* Hypergeometric function of a matrix argument
* Lauricella hypergeometric series
* Modular hypergeometric series
* Riemann's differential equation
* Theta hypergeometric series

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