The Error Is Smaller Than the Term You Left Out

You'll learnto read the error bound off the bracket that produces it, to take the bound from the first term you left out rather than the last one you used, to see why that off-by-one never shows up in a bound you check, to answer a how-many-terms question without losing a term to it, 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 choosing the wrong one still produces a true statement. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · THE FIRST TERM YOU LEFT OUT · v2

The gap to the limit is no bigger than the next term

Before you start

What you're looking at

Dots at the whole numbers along the bottom, each one a running total. The lime rule is the value they are heading for, so the height from a dot up to that rule is the error you would make by stopping there.

The question

Stop adding after four terms and you are off by some amount you cannot compute, because you do not know the answer. How large can that amount possibly be?

Step 1 / 12