Alternating series convergence

What is an alternating series? Does an alternating series always converge? An alternating series is any series , , for which the series terms can be written in one of the following two forms. There are many other ways to deal with the alternating sign , but they can all be written as one of the two forms above. You can say that an alternating series converges if two conditions are met: Its nth term converges to zero.

In mathematical analysis, the alternating series test is the method used to prove that an alternating series with terms that decrease in absolute value is a convergent series. A powerful convergence theorem exists for other alternating series that meet a few conditions. Conditional Convergence is conditionally convergent if converges but does not.

EX Classify as absolutely convergent , conditionally convergent or divergent. Infinite series whose terms alternate in sign are called alternating series. Alternating p-series are detailed at the end.

I Note that an alternating series may converge whilst the sum of the absolute values diverges. In particular the alternating harmonic series above converges. The signs of the general terms alternate between positive and negative. A series in which successive terms have opposite signs is called an alternating series. It’s also known as the Leibniz’s Theorem for alternating series.

However, here is a more elementary proof of the convergence of the alternating harmonic series. Then the alternating series converges. The odd numbered partial sums, , , , and so on, form a non-increasing sequence, because , since. This sequence is bounded below by , so it must converge, say.

Likewise, the partial sums , , , and so on, form a non-decreasing sequence that is bounded above by ,. The integral test and the comparison test given in previous lectures, apply only to series with positive terms. We also have the following fact about absolute convergence. We already know that the series of absolute values does not converge by a previous example. The alternating series test is a simple test we can use to find out whether or not an alternating series converges (settles on a certain number).

In this terminology, the series (.2) converges absolutely while the alternating harmonic series is conditionally convergent. We will also examine the convergence of alternating series by using a method called the alternating series test. The test requires two conditions, which is listed below.

Keep in mind that if you cannot fulfill these conditions, that does not mean the alternating series is divergent. There is still a possibility that it is convergent. This in particular applies to your series. The theorem states that rearranging the terms of an absolutely convergent series does not affect its sum. This implies that perhaps the sum of a conditionally convergent series can change based on the arrangement of terms.

A typical alternating series has the form where for all. We will refer to the factor as the alternating symbol. Some examples of alternating series are. This is an alternating geometric series with.

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.