Alternating series test
Encyclopedia
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
where all the an are non-negative, is called an alternating series. If the limit of the sequence
an approaches 0 as n approaches infinity, and the sequence an is monotone decreasing (i.e. each an is smaller than an−1), then the series converges. If L is the sum of the series,
then the partial sum
approximates L with error
It is perfectly possible for a series to have its partial sums Sk fulfill this last condition without the series being alternating. For a straightforward example, consider:
. The limit of the sequence
equals 0 as 
approaches infinity, and each 
is smaller than 
(i.e. the sequence 
is monotone decreasing).
. As every sum in brackets is non-positive, and as 
, then the (2n+1)-th partial sum is not greater than 
.
That very (2n+1)-th partial sum can be written as
. Every sum in brackets is non-negative. Therefore, the series 
is monotonically increasing: for any 
the following holds: 
.
From the two paragraphs it follows by the monotone convergence theorem that there exists such a number s that
.
As
and as 
, then 
. The sum of the given series is 
, where 
is a finite number. Thus, convergence is proved.
Another way to prove this is showing that the sequence of partial sums are a cauchy sequence.
is monotonically increasing. Since 
, and every term in brackets is non-positive, we see that 
is monotonically decreasing. By the previous paragraph, 
, hence 
. Similarly, since 
is monotonically increasing and converging to 
, we have 
. Hence we have 
for all n.
Therefore if k is odd we have
, and if k is even we have 
.
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
of terms converge. It was discovered by Gottfried Leibniz
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....
and is sometimes known as Leibniz's test or the Leibniz criterion.
A series of the form
where all the an are non-negative, is called an alternating series. If the limit of the sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
an approaches 0 as n approaches infinity, and the sequence an is monotone decreasing (i.e. each an is smaller than an−1), then the series converges. If L is the sum of the series,
then the partial sum
approximates L with error
It is perfectly possible for a series to have its partial sums Sk fulfill this last condition without the series being alternating. For a straightforward example, consider:
Proof
We are given a series of the form. The limit of the sequenceSequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
equals 0 as approaches infinity, and each is smaller than (i.e. the sequence is monotone decreasing).Proof of convergence
The (2n+1)-th partial sum of the given series is. As every sum in brackets is non-positive, and as , then the (2n+1)-th partial sum is not greater than .That very (2n+1)-th partial sum can be written as
. Every sum in brackets is non-negative. Therefore, the series is monotonically increasing: for any the following holds: .From the two paragraphs it follows by the monotone convergence theorem that there exists such a number s that
.As
and as , then . The sum of the given series is , where is a finite number. Thus, convergence is proved.Another way to prove this is showing that the sequence of partial sums are a cauchy sequence.
Proof of partial sum error
In the proof of convergence we saw that is monotonically increasing. Since , and every term in brackets is non-positive, we see that is monotonically decreasing. By the previous paragraph, , hence . Similarly, since is monotonically increasing and converging to , we have . Hence we have for all n.Therefore if k is odd we have
, and if k is even we have .Literature
- Knopp, Konrad, "Infinite Sequences and Series", Dover publications, Inc., New York, 1956. (§ 3.4) ISBN 0-486-60153-6
- Whittaker, E. T., and Watson, G. N., A Course in Modern Analysis, fourth edition, Cambridge University Press, 1963. (§ 2.3) ISBN 0-521-58807-3
- Last, Philip, "Sequences and Series", New Science, Dublin, 1979. (§ 3.4) ISBN 0-286-53154-3