Two Conditions, and a Squeeze That Closes

You'll learnto check both conditions of the alternating series test rather than the obvious one, to see the odd and even partial sums close on each other from opposite sides, to separate a series that converges from one whose absolute values do, to say what the test cannot conclude when it fails, and to read the error bound straight off the same squeeze that gave the convergence.

Alternating signs let the partial sums overshoot and correct, closing in from both sides at once. That is enough to rescue a series whose absolute values have no hope: the alternating harmonic series converges to the natural logarithm of 2 while the harmonic series diverges. But it takes two conditions, and only one of them is usually checked.

8 STEPS · 6 QUICK CHECKS · TWO CONDITIONS, AND ONE IS USUALLY SKIPPED · v1

eventually decreasing · tending to zero · both, or nothing

Before you start

What you're looking at

An index panel. Pink stems are the terms, and in this topic they stand above and hang below the axis, because the signs alternate. Violet dots are the odd partial sums, teal dots the even ones, and the lime rule is the limit they close on.

Two sequences, not one

From step 2 the partial sums are drawn as two separate interleaved families rather than one zigzag. The odd ones fall, the even ones rise, and every odd one stays above every even one — that is the squeeze, and it is what the test is really about.

Watch for

In step 3 a sequence tends to zero while rising at every other step. It passes one condition and fails the other, and the test says nothing at all about it — which is the case that shows why both conditions are stated separately.

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