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In mathematics, an alternating series is an infinite series of the form
with an > 0
for all n
. The signs of the general terms alternate between positive and negative. Like any series, an alternating series converges
if and only if the associated sequence of partial sums converges
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 Mercator series provides an analytic expression of the natural logarithm:
The functions sine and cosine used in trigonometry can be defined as alternating series in calculus even though they are introduced in elementary algebra as the ratio of sides of a right triangle. In fact,
When the alternating factor (–1)n
is removed from these series one obtains the hyperbolic functions
sinh and cosh used in calculus.
For integer or positive index α the Bessel function of the first kind may be defined with the alternating series
is the gamma function
If s is a complex number, the Dirichlet eta function is formed as an alternating series
that is used in analytic number theory
Alternating series test
The theorem known as "Leibniz Test" or the alternating series test tells us that an alternating series will converge if the terms an converge to 0 monotonically.
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: . Similarly, it can be shown that . Since converges to , our partial sums form a Cauchy sequence (i.e., the series satisfies the Cauchy criterion) and therefore converge. The argument for even is similar.
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:
That does not mean that this always finds the very first element after which error is the modulus of the next term in the series. Indeed if you will take
and will try to find the term after which error is 0.00005, you will find the term that is 2 times more than optimal term. Indeed, the error after summing from first 9999 elements is 0.0000500025, and the next term is the one. This series happens to have a property that constructing a new series with
also gives an alternating series where Leibniz test applies and thus makes this simple error bound not optimal. This requires discovered in 1962 Calabrese bound, that this property gives result 2 times less than with Leibniz error bond. In fact this is also not optimal for series where this property applied 2 or more times, which is described by Johnsonbaugh
error bond. In fact, if one can apply the property infinite ammount of times, Euler transform series
A series converges absolutely 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. Therefore, the series converges as the difference of two convergent series .
A series is conditionally convergent if it converges but does not converge absolutely.
For example, the harmonic series
diverges, while the alternating version
converges by the alternating series test
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 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
But, since the series does not converge absolutely, we can rearrange the terms to obtain a series for :
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.