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Taylor Series Calculator

Type any f(x) — tan(x), sqrt(1+x), ln(1+x), exp(x)/cos(x) — pick the expansion point and get exact rational coefficients up to order 20, the derivatives f⁽ⁿ⁾(a) and the detected general term.

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Taylor Series Calculator is free to use as often as you like, directly from this page. Its place is under Calculus; Fourier Series Coefficients Calculator and Maclaurin Series Calculator answer the questions closest to this one.

How to use it

  1. Open the tool — no signup or install needed.
  2. Enter your input or adjust the available options.
  3. Get your result instantly, then copy or download it.

Frequently asked questions

What is Taylor Series Calculator?

Type any f(x) — tan(x), sqrt(1+x), ln(1+x), exp(x)/cos(x) — pick the expansion point and get exact rational coefficients up to order 20, the derivatives f⁽ⁿ⁾(a) and the detected general term.

When would I actually use this?

Checking a derivative or an integral you worked out by hand, finding where a function turns, and getting a numeric answer when no closed form exists.

What is the most common mistake?

Integrating across a discontinuity as if it were not there. A numeric method will happily return a finite value for an integral that diverges — check the domain before trusting the number.

How is Taylor Series Calculator different from Fourier Series Coefficients Calculator?

They sit next to each other but answer different questions. Fourier Series Coefficients Calculator: a₀, aₙ and bₙ over any period, computed exactly by integration by parts when f is a polynomial and by Gauss–Legendre quadrature otherwise, with the truncated series plotted against f so the Gibbs overshoot is visible. Pick whichever matches what you're starting from — both are free.

Is there a tool for the next step?

Maclaurin Series Calculator is the closest one after this: The Taylor series at 0 for twelve standard functions — sin, cos, eˣ, ln(1+x), √(1+x), arcsin, arctan and more — with exact fractions such as 1/120, the general term and the error at a chosen x.

What else is worth having open alongside it?

Infinite Series Sum Calculator and Power Series Calculator — they come up in the same task often enough to be worth a second tab.

Where do the figures come from?

Symbolic results are exact; numeric ones come from adaptive quadrature or a standard step method, and the tool reports which. Where both are available, compare them — a large gap means the problem is ill-conditioned.

Further reading

All guides →
ExplainerSprings in Series Behave Like Resistors in ParallelTwo 500 N/m springs end to end give 250 N/m; side by side they give 1,000. Under the same 50 N load that is 0.2 m of travel against 0.05 m, and 5 joules stored against 1.25.ExplainerCapacitors in Series: the Smallest One Decides, and Then It Fails100, 220 and 330 µF in series come to 56.9 µF — less than the smallest. Put 100 µF and 1 µF in series and the total is 0.99 µF, while the small one carries 99 % of the voltage.ExplainerResistors in Series and in Parallel: The Formulas, and the Three Things They Do Not Tell YouSeries adds, parallel adds reciprocals — that part is easy. The parts that catch people: which member dominates the answer, why two 5 % resistors in parallel are still 5 %, and why the smallest resistor in a parallel pair takes the largest share of the power.How-toHow to Calculate an LED Series ResistorSize the series resistor for any LED with one formula: R = (Vs − Vf) / I. Follow the steps, see a worked 5 V example, and pick the right resistor.ExplainerThe Pigeonhole Principle Proves a Thing Exists Without Finding ItTen items in three containers force four into one of them. 367 people force a shared birthday. Neither statement tells you which container or which pair.ExplainerA Cone Is Exactly One Third of Its CylinderRadius 5, height 10: the cone holds 261.8 and the cylinder around it holds 785.4. The ratio is exactly a third, and it holds for any base shape at all.