Taylor remainder bound is one of 6 series formulas in the calculus section of this library, and it is used at ap · university level.
Why taylor remainder bound works
The Lagrange form says the leftover after a degree-n polynomial equals the next derivative, measured at some unknown point between the centre and the input, times the next power over the next factorial. You cannot find that point, so replacing the derivative with any ceiling that holds throughout gives an honest limit on the error.
What each symbol means
$M$ bounds $|f^{(n+1)}|$ between $a$ and $x$.
Taylor remainder bound: when it holds
A valid finite bound $M$ must hold throughout the interval between center and input.
When it stops applying
If the next derivative is unbounded between the centre and the input, no finite ceiling exists and the estimate says nothing at all. That is what happens near a point where the function or one of its derivatives blows up.
Taylor remainder bound: a worked example
For $e^x$ near $0$, choose $M=e^{|x|}$ to bound the next-order error.
The mistake to avoid
What people do: Using the size of the next derivative at the centre instead of a ceiling across the whole stretch.
Why it goes wrong: The unknown point can be anywhere between the centre and the input, where that derivative may be much larger, so the so-called bound can end up smaller than the actual error.
Do this instead: Take the largest size the next derivative reaches anywhere between the centre and the input.
Taylor remainder bound: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $M$ bounds $|f^{(n+1)}|$ between $a$ and $x$.
- Check the conditions before substituting. A valid finite bound $M$ must hold throughout the interval between center and input.
- 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 taylor remainder 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.
- Taylor Series 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 taylor remainder bound
Does a small bound mean my answer is that accurate?
It means the error is no worse than the bound. The real error is often far smaller, because the ceiling describes the worst case over the whole stretch.
Which derivative am I supposed to bound?
The one whose order is one higher than the degree you kept. After stopping at degree n, the first term you discarded involves the derivative of order n+1.
Why does the bound shrink so quickly?
The factorial in the denominator eventually outgrows any fixed power in the numerator, so for inputs near the centre the estimate collapses fast as terms are added.
Should I use this for an alternating series?
You can, but the alternating-series estimate is usually simpler and tighter, because there the error is capped by the size of the first term you left out.