What a Taylor Series Is
A Taylor series rebuilds a function from its derivatives at a single point: f(x) = Σ f(n)(a)/n! · (x − a)n. When the centre is a = 0 it is called a Maclaurin series. Each extra term corrects the curve a little more.
Brook Taylor published the general formula in 1715, and Colin Maclaurin popularised the a = 0 case in 1742. Your calculator’s sin and exp keys run on polynomials built this way.
Maclaurin series of eˣ
- Every derivative of eˣ is eˣ, and at 0 each equals 1.
- So the coefficients are 1/n!.
Answer: 1 + x + x²/2! + x³/3! + x⁴/4! + …
The Series to Memorise
| Function | Maclaurin series | Converges for |
|---|---|---|
| eˣ | 1 + x + x²/2! + x³/3! + … | all x |
| sin x | x − x³/3! + x⁵/5! − … | all x |
| cos x | 1 − x²/2! + x⁴/4! − … | all x |
| 1/(1 − x) | 1 + x + x² + x³ + … | |x| < 1 |
| ln(1 + x) | x − x²/2 + x³/3 − … | −1 < x ≤ 1 |
| arctan x | x − x³/3 + x⁵/5 − … | |x| ≤ 1 |
Substituting is faster than recomputing. The series for e−x² comes from eˣ by replacing x with −x²: 1 − x² + x⁴/2 − x⁶/6 + …
How Accurate Is a Taylor Polynomial?
Use four terms (up to x³)
- 1 + 0.1 + 0.005 + 0.000166667 = 1.105166667.
- True value: 1.105170918.
- Error: about 4.3 × 10⁻⁶.
- Lagrange bound: |R3| ≤ M·x⁴/4!, with M = e0.1 ≈ 1.105, which gives 1.105 × 0.0001 ÷ 24 ≈ 4.6 × 10⁻⁶.
Answer: e0.1 ≈ 1.10517, accurate to five decimals
Use terms up to x⁵
- 0.5 − 0.125/6 + 0.03125/120.
- 0.5 − 0.0208333 + 0.0002604 = 0.4794271.
- True value 0.4794255.
Answer: sin 0.5 ≈ 0.47943
The error bound is the answer to “how many terms do I need?” Pick the target error, then raise n until the bound drops below it.
Taylor Series About Other Centres
Choose the centre near where you need the values. ln x cannot be expanded about 0, but about a = 1 it works.
Estimate ln 1.2
- ln x = (x−1) − (x−1)²/2 + (x−1)³/3 − …
- With x − 1 = 0.2: 0.2 − 0.02 + 0.002667 − 0.0004.
- Sum: 0.182267. True value 0.182322.
Answer: ln 1.2 ≈ 0.1823
In physics the first-order term turns sin θ into θ. A pendulum at 10° (0.1745 rad) has sin θ = 0.17365, an error of 0.5% from using θ. That is why small-swing pendulums obey a simple formula.
What We Think Is Overrated, Underrated and Wrong
Original Data: How Fast Do the Terms Converge?
| Terms of eˣ at x = 1 | Partial sum | Error vs e = 2.718282 |
|---|---|---|
| 2 terms | 2.00000 | 0.718 |
| 4 terms | 2.66667 | 0.0516 |
| 6 terms | 2.71667 | 0.00161 |
| 8 terms | 2.71825 | 0.0000028 |
For sin x ≈ x − x³/6 at x = 0.5 the approximation is 0.479167 and the true value is 0.479426, an error of about 0.00026. Near the centre, a few terms are enough.
Series to Know
| Function | Series about 0 | Radius |
|---|---|---|
| eˣ | 1 + x + x²/2! + x³/3! + … | ∞ |
| sin x | x − x³/3! + x⁵/5! − … | ∞ |
| cos x | 1 − x²/2! + x⁴/4! − … | ∞ |
| 1/(1 − x) | 1 + x + x² + … | 1 |
| ln(1 + x) | x − x²/2 + x³/3 − … | 1 |
Our Take: What Is Overrated, Underrated and Plain Wrong
Overrated: memorising long series. Know five and derive the rest by substitution or differentiation.
Underrated: the radius of convergence. A series can be perfect inside it and useless outside.
Plain wrong: using ln(1 + x) at x = 2. The series diverges there (radius 1), even though ln 3 is a fine number.
Taylor Series Mistakes That Cost Marks
- Forgetting the n! in the denominator.
- Using the wrong centre: (x − a)ⁿ, not xⁿ.
- Dropping alternating signs in sin and cos.
- Using the series outside its radius of convergence.
- Confusing the degree with the number of terms.
Why Students Get Stuck on Taylor Series, and What a Good Calculator Should Do
Searching for help with taylor series usually starts late at night with one problem that will not work out. The formula uses derivatives at a point, and the pattern is easy to lose. A good calculator fixes that by showing each step, checking its own answer and explaining the idea in plain words. This section covers what students struggle with, what they want, and how this page responds.
In short: students need to build the polynomial term by term and estimate the error, not just see a final number, and a calculator earns trust only when its steps can be checked.
Understand the pain points
What students get stuck on
Most mistakes here are not about arithmetic. They come from one rule applied the wrong way round, which is why seeing every step matters.
- Forgetting the factorial in the denominatorTaylor series of e^x at 0 up to degree 4↗
- Taking derivatives at the wrong pointTaylor series of sin(x) at 0 up to degree 5↗
- Not seeing the pattern of termsTaylor series of ln(1 + x) at 0 up to degree 4↗
- Not knowing how good the approximation isTaylor series of cos(x) at 0 up to degree 4↗
Understand the frustrations and desires
What they feel, and what they want
Each frustration points to a wish: a clearer step, a built-in check, an explanation in plain words.
- Wants each derivative listed
- Wants the polynomial written out
- Wants a feel for the error
Build the page around their intent
Each intent has a place on this page
The page follows intent, not features, so each need has a clear starting point.
- I need an answer now: type the problem into the calculator at the top of this page.Taylor series of e^x at 0 up to degree 4↗
- I need to understand: the sections below explain the idea in plain words and list the common mistakes.
- I need to check my work: every step is shown, so the first line that differs from yours is the one to study.
- I need practice: open the solver, tap + and choose Create Practice Test.
- I need it explained aloud: in the solver, tap + and choose Create Video.