Taylor Series Calculator
Expand a function as a Taylor or Maclaurin series about any centre. Get the polynomial, the coefficients, the truncation error and the radius of convergence.
Taylor polynomial — 6 terms
P(x) = 1 + x + 0.5x^2 + 0.16666667x^3 + 0.04166667x^4 + 0.00833333x^5
Coefficient rule used: f⁽ᵏ⁾(a) = eᵃ, so cₖ = eᵃ / k!
Coefficients
| k | cₖ = f⁽ᵏ⁾(a)/k! | term |
|---|---|---|
| 0 | 1 | 1 |
| 1 | 1 | 1x |
| 2 | 0.5 | 0.5x^2 |
| 3 | 0.1666666667 | 0.16666667x^3 |
| 4 | 0.0416666667 | 0.04166667x^4 |
| 5 | 0.0083333333 | 0.00833333x^5 |
Radius of convergence: infinite — the series converges for every x.
What is the Taylor Series Calculator?
The ByteTools Taylor Series Calculator expands the standard functions — eˣ, sin, cos, ln(1 + x), 1/(1 − x), √(1 + x) and arctan — from their closed-form coefficient rules, so the coefficients are exact rather than approximated by repeated numerical differentiation.
- Closed-form coefficient rules for seven standard functions
- Any centre a, not just Maclaurin series about zero
- Exact expansion of any polynomial you type
- Truncation error shown against the true value at a test point
- Radius of convergence stated, with a warning outside it
- All computed locally in your browser
How to use the Taylor Series Calculator
- 1
Pick a function from the list, or choose 'Custom polynomial in x' and type your own.
- 2
Set the centre a — leave it at 0 for a Maclaurin series.
- 3
Choose how many terms you want and the x value to test the polynomial at.
- 4
Read the partial-sum polynomial, then compare its value against the true value and the truncation error.
- 5
Check the coefficient table and the stated radius of convergence, then copy the series.
About the Taylor Series Calculator
The ByteTools Taylor Series Calculator expands the standard functions — eˣ, sin, cos, ln(1 + x), 1/(1 − x), √(1 + x) and arctan — from their closed-form coefficient rules, so the coefficients are exact rather than approximated by repeated numerical differentiation. You choose the centre, the number of terms and a test point.
It prints the partial-sum polynomial, lists every coefficient cₖ = f⁽ᵏ⁾(a)/k!, evaluates the polynomial at your test point and shows the truncation error against the true value. The radius of convergence is stated, and the page warns you when your test point falls outside it — where a partial sum is not an approximation of anything.
You can also expand any polynomial you type, exactly, by repeated symbolic differentiation. Everything runs in your browser and nothing is uploaded.
Frequently asked questions
What is the difference between a Taylor series and a Maclaurin series?
A Maclaurin series is simply a Taylor series centred at a = 0. Any other centre makes it a general Taylor series. This calculator handles both — leave the centre at zero for a Maclaurin expansion.
How many terms of a Taylor series do I need?
It depends on how far your point is from the centre and how much accuracy you want. The truncation error shown on this page tells you directly: add terms until that error is small enough for your purposes.
What is the radius of convergence?
It is the distance from the centre within which the infinite series actually converges to the function. For eˣ, sin and cos it is infinite. For ln(1 + x) and 1/(1 − x) it is the distance from the centre to the point where the function breaks down.
Why does the calculator only accept polynomials for custom functions?
Building coefficients for an arbitrary function needs repeated symbolic differentiation, and for anything involving division or a chain of functions the expression grows explosively after a handful of derivatives. Polynomials shrink instead, so they stay exact and fast.
Why is arctan only expanded about zero?
Unlike the other functions here, arctan has no tidy closed-form expression for its kth derivative at a general point. Rather than return a poorly conditioned numerical guess, the tool offers it as a Maclaurin series and says so.
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