What is the purpose of alternating series test?

What is the purpose of alternating series test?

The alternating series test (also known as the Leibniz test), is type of series test used to determine the convergence of series that alternate. Keep in mind that the test does not tell whether the series diverges. In order to use this test, we first need to know what a converging series and a diverging series is.

What does the alternating series test say?

The alternating series test can only tell you that an alternating series itself converges. The test says nothing about the positive-term series. In other words, the test cannot tell you whether a series is absolutely convergent or conditionally convergent.

What are the two conditions for convergence of an alternating series test?

In other words, if the absolute values of the terms of an alternating series are non-increasing and converge to zero, the series converges.

Can an alternating series be monotonic?

For the convergence of an alternating series, the sequence {pn} needs to be a non-negative, monotonically decreasing sequence with a limit of zero. A non-negative sequence with limit zero whose alternating series diverges.

Are alternating series decreasing?

The series from the previous example is sometimes called the Alternating Harmonic Series. In the previous example it was easy to see that the series terms decreased since increasing n only increased the denominator for the term and hence made the term smaller.

What happens if the alternating series test fails?

If the alternating series fails to satisfy the second requirement of the alternating series test, it does not follow that your series diverges, only that this test fails to show convergence.

What is alternating series error bound?

The error is the difference between any partial sum and the limiting value, but by adding an additional term the next partial sum will go past the actual value. Thus for a convergent alternating series the error is less than the absolute value of the first omitted term: .

What defines an alternating series?

In mathematics, an alternating series is an infinite series of the form or. 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.

Do alternating series have a limit?

Given an alternating series ∑(−1)kak, ∑ ( − 1 ) k a k , if the sequence {ak} of positive terms decreases to 0 as k→∞, k → ∞ , then the alternating series converges. Note that if the limit of the sequence {ak} is not 0, then the alternating series diverges.

How do you show a sequence is monotonically increasing?

A sequence (an) is monotonic increasing if an+1≥ an for all n ∈ N. The sequence is strictly monotonic increasing if we have > in the definition. Monotonic decreasing sequences are defined similarly. A bounded monotonic increasing sequence is convergent.

What is the alternating series test?

The alternating series test (also known as the Leibniz test), is type of series test used to determine the convergence of series that alternate. Keep in mind that the test does not tell whether the series diverges. In order to use this test, we first need to know what a converging series and a diverging series is.

How do you check if an alternating series diverges?

So here is a good way of testing a given alternating series: if you see the alternating series, check first the nth Term Test for Divergence (i.e., check if lim. n!1. (¡1)n¯1u. n does not exist or converge to a non-zero value). If this test holds, then the series diverges and it’s the end of the story.

What is an alternating harmonic series?

The series from the previous example is sometimes called the Alternating Harmonic Series. Also, the (−1)n+1 ( − 1) n + 1 could be (−1)n ( − 1) n or any other form of alternating sign and we’d still call it an Alternating Harmonic Series.