Alternating-series error bound
Bound the error after stopping a convergent alternating series.
Alternating-series error bound is one of 6 series formulas in the calculus section of this library, and it is used at ap · university level.
Why alternating-series error bound works
When the terms alternate and steadily shrink, each partial sum overshoots the true value and the next one undershoots it, so the answer is always trapped between two consecutive partial sums. The gap between those two is the size of the very next term, which is why the error can never exceed it.
What each symbol means
$b_n$ is the positive magnitude of the $n$th alternating term.
Alternating-series error bound: when it holds
The magnitudes must decrease and approach zero.
When it stops applying
The sizes must decrease steadily all the way down to zero. If they only shrink on average, jumping back up now and then, the neat trapping between consecutive partial sums collapses and the next term is no longer a valid ceiling.
Alternating-series error bound: a worked example
For $1-1/2+1/3-\cdots$, the error after $n$ terms is at most $1/(n+1)$.
The mistake to avoid
What people do: Quoting the size of the last term that was added rather than the first one left out.
Why it goes wrong: The bound comes from the term that would come next, so naming an included term reports a larger error than the argument actually supports.
Do this instead: Count carefully: after adding a certain number of terms, the bound is the size of the term with the next index.
Alternating-series error bound: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $b_n$ is the positive magnitude of the $n$th alternating term.
- Check the conditions before substituting. The magnitudes must decrease and approach zero.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Series
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where alternating-series error bound comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Sequences and Series — the lesson behind this formula: recognize arithmetic, geometric, and recursive patterns.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about alternating-series error bound
Does this bound work for every alternating series?
Only for those whose term sizes decrease to zero. Without that the series may not converge at all, so there is nothing to put a bound on.
How many terms give three-decimal accuracy?
Enough that the next term is under 0.0005. For the alternating harmonic series that means about 2000 terms, which shows how slowly it settles.
Is the real error close to the bound?
It is often roughly half of it, because the true value sits somewhere between two consecutive partial sums rather than at one end of that gap.
Can I use it on a Taylor polynomial?
Yes, whenever the series alternates at that input, as the sine series does for a positive input, and it is usually tighter than the general remainder bound.