Taylor / Maclaurin Series Calculator
Expand a function as a polynomial about point a. Visualize convergence as n increases.
Taylor polynomial P_5(x) ≈ sin(x) about x = 0:
x - 0.1667x^3 + 0.0083x^5
General term: (-1)^n · x^(2n+1) / (2n+1)!
Radius of convergence: ∞
Convergence plot — blue: f(x), dashed orange: P_5(x)
Increase n with the slider to watch the polynomial converge to f(x).
Common Maclaurin Series Reference
| Function | General Term | Radius |
|---|---|---|
| sin(x) | Σ (-1)ⁿ x^(2n+1)/(2n+1)! | ∞ |
| cos(x) | Σ (-1)ⁿ x^(2n)/(2n)! | ∞ |
| eˣ | Σ x^n/n! | ∞ |
| ln(1+x) | Σ (-1)^(n+1) x^n/n (n≥1) | |x|<1 |
| 1/(1-x) | Σ xⁿ | |x|<1 |
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Taylor Series Learning
These walkthroughs cover what Taylor polynomials approximate, how derivative information builds the series, and where that local approximation can break down.
- Find a Taylor polynomial and estimate the error
This example builds the degree-5 Taylor polynomial of $\sin(x)$ centred at $0$, evaluates it at $x=0.5$, and confirms that the actual error stays inside the bound given by Taylor's theorem.
- What a Taylor polynomial is approximating
A Taylor polynomial is not trying to copy a function everywhere. It matches the function and several of its derivatives at one chosen center, so it is designed to be locally accurate near that point.
- Why continuity and differentiability are different concepts
Differentiability is stronger than continuity. A graph can be unbroken at a point and still fail to have a derivative there if the local shape has a corner, cusp, or vertical tangent.