Alternating Series | Research & Encyclopedia Articles

Alternating Series

Alternating series—an infinite series whose terms alternate sign, i.e. whose terms are alternately positive and negative. Examples are the series 1 - 1 + 1 - 1 + 1 ... and the series 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ... , which is called the alternating harmonic series.

In general, it is more difficult to determine whether a series with terms of varying sign converges or diverges than it is with a series whose terms are all of the same sign. For a series to converge, its sequence of partial sums (that is, the sequence obtained by taking just the first term, then taking the sum of the first two terms, then the sum of the first three terms, and so on) must have a limit. If all the terms in a series are positive, the partial sums will all be positive and will get larger and larger as more terms of the series are added on. Therefore there are only two possibilities: either the partial sums will eventually level off, in which case the series converges, or they will grow without bound, in which case the series diverges. However, if the terms are not all positive, there are more possibilities. For instance, with the series of alternating 1's and -1's above, if we group the numbers in pairs, we get (1 - 1) + (1 - 1) + (1 - 1) + ... , which would lead us to think the series converges to 0. On the other hand, if we group them in pairs starting with the second term, we get 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + ... , giving us a sum of 1. To resolve this inconsistency, we look at the sequence of partial sums:

1

1-1=0

1-1+1=1

1-1+1-1=0

1-1+1-1+1=1

and so on. Since the sequence of partial sums does not have a unique limit, the series diverges. Thus an alternating series can diverge even though its partial sums do not grow unboundedly large.

There is one test available to determine if an alternating series converges, called, unsurprisingly, the Alternating Series Test. An infinite series ((-1) ^k+1a_k with a_k > 0 for all k (i.e. the alternating infinite sum a₁ - a₂ + a₃ - a₄...) converges if

• a₁ > a₂ > a₃ > ... and
• lim _{k ( 0} a_k = 0.

Thus, for example, the alternating harmonic series 1 - 1/2 + 1/3 - 1/4 + ... converges. Note that the theorem does not say that a series without the two properties will diverge. (It is true that any series, alternating or not, will diverge if it does not satisfy the second property, but a series can converge without satisfying the first property.)

The alternating harmonic series is an example of a conditionally convergent series, that is, a series that converges but which would not converge if all the negative terms were changed to positive. In other words, the alternating harmonic series is conditionally convergent because it converges but the series 1 + 1/2 + 1/3 + 1/4 ... diverges. A series that does converge if all the negative terms are changed to positive is said to be absolutely convergent. For example, the series 1 - 1/2 + 1/4 - 1/8 + 1/16 - 1/32 + ... is absolutely convergent. Conditionally convergent series have a very interesting property, which has to do with how their terms can be rearranged. A rearrangement of a series has the same numbers being added, but in a different order, and therefore has a different sequence of partial sums. Consider the following rearrangement of the alternating harmonic series (which converges to the natural logarithm of 2, or about .69). First we take just enough positive terms so that the partial sum exceeds 1.2 (a number chosen at random). So our rearrangement starts 1 + 1/3 = 1.33333.... Now we add on just enough negative terms so that the partial sum falls below 1.2:

1 + 1/3 + -1/2 = .833333....

Now we add on positive terms to get the partial sum above 1.2:

1 + 1/3 + -1/2 + 1/5 + 1/7 + 1/9 = 1.28730....

Now negative terms, to get the partial sum below 1.2:

1 + 1/3 + -1/2 + 1/5 + 1/7 + 1/9 - 1/4 = 1.03730....

Continuing in this fashion, we obtain the following partial sums (rounded off):

• 1.20513, 1.03847, 1.21659, 1.09159, 1.22269, 1.12269, 1.22646, 1.14313, ... .

It is not hard to see that these partial sums will approach 1.2 — since the numbers we are adding on are shrinking in size, the amounts by which we "overshoot" 1.2 as we add on positive and negative terms shrinks as well.

Thus we have taken a series that converges to approximately .69 and rearranged it into a series that converges to 1.2. It is easy to see that we could just as easily have rearranged it to give any other number of our choosing. In fact, we can even rearrange it to get a series that diverges: first add enough positive terms to get above 10, say, and then enough negative terms to get below -10, then enough positive terms to get above 10 again, then enough negative terms to get below -10, and so on. Thus, against all our intuition, we have shown that addition can sometimes fail to be commutative once we shift from the finite addition we learned in grade school to the addition of infinite collections of numbers.

This is the complete article, containing 902 words (approx. 3 pages at 300 words per page).