Two Conditions, and a Squeeze That Closes

You'll learnto check both conditions of the alternating series test rather than the obvious one, to watch the odd and even totals close on each other from opposite sides, to separate a series that converges from one whose absolute values do, and to read the error bound straight off the same squeeze that gave the convergence.

Alternating signs let the running total 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. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · TWO CONDITIONS, AND ONE IS USUALLY SKIPPED · v2

Eventually falling, and heading to zero: both, or no conclusion

Before you start

What you're looking at

An index panel. Pink stems are the terms, standing above the axis and hanging below it in turn. Violet dots are the odd running totals and teal dots the even ones.

The question

If the total keeps jumping past the answer and back, how do you know there is an answer?

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