Sequences and Series, Alternating Series

Alternating Series

After some leading terms, the terms of an alternating series alternate in sign, approach 0, and never increase in absolute value. (This is a series of real numbers.) We will show that such a series converges. An example is 1-1/2+1/3-1/4+1/5-1/6…, which approaches log(2).

To keep things simple, set aside any leading terms, so the entire series meets the above criteria. Let s1, the first term, = w, where w is positive.

Group terms two by two, and we are constantly adding small positive values to the partial sum. In other words, s1+s2 is positive, or perhaps 0, and the same for s3+s4, and s5+s6, and so on. The partial sums of even index form a monotonically increasing sequence.

At the same time, the odd partial sums are w, w+s2+s3, w+s2+s3+s4+s5, and so on. These odd partial sums form a monotonically decreasing sequence.

Again, we have an increasing sequence of partial sums that starts at 0, and a decreasing sequence of partial sums that starts at w. If the increasing sequence ever exceeds w, add the next term of s, which is positive, and find a member of the decreasing sequence that is also larger than w. Yet the decreasing sequence starts at w and descends from there. Thus the increasing sequence is bounded by w and converges to something between 0 and w. At the same time, the decreasing sequence cannot drop below 0, hence it converges to something between 0 and w.

As a generalization of the above, the increasing sequence can never go above the decreasing sequence. If it does, bring in the next term of s and find a new partial sum that is part of the decreasing sequence, yet it lies above the decreasing sequence at that point. Therefoore the limit of the increasing sequence is no larger than the limit of the decreasing sequence.

Suppose the two limits are not equal. If they differ by ε then go out beyond sn, where terms are smaller than ε. Now the terms in the increasing and decreasing sequence must be within ε of each other. This is a contradiction, therefore both sequences approach the same limit, the limit of s. every alternating sum converges.