The Error Is Smaller Than the Term You Left Out

You'll learnto read the error bound off the bracket that produced it, to take the bound from the first omitted term rather than the last one used, to see why that off-by-one is never caught by checking a bound, to answer the how-many-terms question without losing a term to it, to tell this bound apart from the Lagrange bound by the machinery each needs, and to know what breaks when the terms stop decreasing.

Alternating series come with something no other convergent series gives you: a guaranteed error, available without knowing the sum. Stop after some number of terms and you are within the next term of the answer. The bound is that simple, and the whole difficulty is which term counts as next — because getting it wrong produces a statement that is still true.

8 STEPS · 6 QUICK CHECKS · THE FIRST TERM YOU LEFT OUT · v1

the gap to the limit is smaller than one term

Before you start

What you're looking at

An index panel. The horizontal axis is n and takes whole-number values only. The vertical axis is a running total in most steps, so a lime rule is the value the totals are heading for and the vertical distance from a dot to that rule is the error at that index.

Three hues, three quantities

Pink is the approximation and the error it actually makes. Teal is the bound the theorem gives — the first term you left out. Violet is a rival bound: the last term you used in steps 3 to 5, and the Lagrange remainder in step 6. A bound drawn in red is one that fails.

Watch for

In step 2 the teal segment reaches past the lime rule. That overshoot is not sloppiness — it is what a bound is. In step 7 the same segment stops short of the rule, and that is the whole difference between a bound that is loose and one that is false.

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