Interactive guide
Every smooth curve has a polynomial hiding inside it.
The true curve (blue) versus the polynomial built from N terms (orange).
true function Taylor polynomial radius of convergence
1
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- Error at x = 2.0
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Watch the orange curve near x = 0: with just one or two terms it already traces the blue curve almost exactly close to the center. Look further out, toward the edges of the chart, and the same polynomial peels away, sometimes gently, sometimes explosively. More terms buys you a wider well-behaved region, not a perfect copy everywhere at once.
Matching slopes, not just heights
A Taylor polynomial isn't just close to f(x) at the center,
it's built to match f's value, its slope, its curvature, and
increasingly many derivatives beyond that, all at a single
point. Term k is f(k)(0)/k! ×
xk, literally "the k-th derivative, packaged
so that adding this term fixes the k-th derivative's match and
leaves every earlier one exactly as it was."
That's a purely local promise. Nothing about matching more and more derivatives at x = 0 guarantees anything at all far away from 0, which is exactly why the last preset below breaks so differently from the first three.
Reading the chart
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1
sin, cos, eˣ: converge everywhere
Push N high enough and the orange curve eventually matches the blue one across the entire visible window.
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2
1/(1−x): only within the shaded band
Inside |x| < 1 more terms genuinely help. Outside it, more terms make the polynomial worse, not better, no matter how high N goes.
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3
That boundary is the radius of convergence
A fixed distance from the center, past which the infinite series simply doesn't add up to anything.
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4
sin and cos trade off which half improves
Each new term is a higher odd or even power, so it bends the far tails of the curve more than the middle, which was already settled.
This is the same "match the slope at a point" idea from the Derivatives & Integration page, run over and over: first match f itself, then match how fast it's changing, then how that rate is changing, and so on. A tangent line is a Taylor polynomial with exactly one term (N = 1) beyond the constant, this page is just that same construction, continued.
The jargons
One recipe, four functions, one boundary where it stops working.
An infinite sum of terms built from a function's derivatives at one point, designed to reproduce the function near that point.
A Taylor series centered at x = 0, the case used throughout this page.
The polynomial you get by stopping after N terms instead of summing forever, exactly what the slider controls.
The distance from the center within which the infinite sum settles on a finite value. Outside it, adding more terms never helps.
The result of differentiating f a total of k times; each term's coefficient comes from exactly one of these, evaluated at the center.
1×2×3×...×k. It grows fast enough to shrink later terms' coefficients toward zero, which is what lets the sum converge at all.