Taylor series
Represent a sufficiently regular function by derivatives at a center.
Taylor series is one of 6 series formulas in the calculus section of this library, and it is used at ap · university level.
Why taylor series works
The coefficients are forced on you. If a power series really equals the function near the centre, then differentiating n times and substituting the centre wipes out every term but one, and that survivor carries a factor of n factorial. Dividing by that factorial recovers the coefficient.
What each symbol means
$a$ is center and $f^{(n)}(a)$ the order-$n$ derivative there.
Taylor series: when it holds
The Taylor series must converge to $f$ at the chosen $x$; smoothness alone is not always enough.
When it stops applying
Having every derivative is not enough to guarantee the series returns the function. The function equal to e to the power of −1 over x squared away from the origin, and 0 at the origin, has every derivative equal to 0 there, so its series is identically 0 while the function is not.
Taylor series: a worked example
$e^x=1+x+x^2/2!+x^3/3!+\cdots$ around $a=0$.
The mistake to avoid
What people do: Leaving out the factorial and using the raw derivative as the coefficient.
Why it goes wrong: Every differentiation of a power term leaves a factorial behind, so without dividing it out the terms are enormously too large and the sum does not match the function.
Do this instead: Divide each derivative by the factorial of its own order before attaching the matching power.
Taylor series: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $a$ is center and $f^{(n)}(a)$ the order-$n$ derivative there.
- Check the conditions before substituting. The Taylor series must converge to $f$ at the chosen $x$; smoothness alone is not always enough.
- 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 series 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 series
What makes a series a Maclaurin series?
Only the choice of centre. A Maclaurin series is a Taylor series centred at 0, and nothing else about the construction changes.
How many terms should I keep?
Enough that the remainder estimate falls under whatever error you are allowed. The nearer the input is to the centre, the fewer terms that takes.
Why does the series stop working far from the centre?
Beyond the radius of convergence the terms stop shrinking, so the partial sums wander off instead of settling, no matter how many terms you add.
How is this different from a tangent-line estimate?
A tangent-line estimate is this series cut off after the first-degree term. Keeping the second-degree term as well brings the curvature of the graph into the approximation.