Skip to content
Allin
Sign up · Sign in

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.

Related tools

All Calculus tools →
Fourier Series Coefficients Calculatora₀, 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.Maclaurin Series CalculatorThe 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.Infinite Series Sum CalculatorTen classic families — geometric, p-series, telescoping, alternating harmonic, Leibniz, Basel, Σ1/n! — with their closed form (π²/6, ln 2, π/4, e), the convergence condition and the partial sums.Power Series CalculatorExpand eˣ, sin, cos, ln, √x, arctan and six more about any centre a, with exact fractional coefficients, the general term, and the interval of convergence including both endpoints.Series Convergence Test CalculatorPick a series and see which test settles it — p-series, ratio, root, Leibniz, Cauchy condensation, limit comparison — with the limit it produces and whether the convergence is absolute or conditional.Antilog Calculator (inverse logarithm)The antilogarithm undoes the logarithm: antilog_b(y) = bʸ, so log_b(bʸ) = y. Pick base 10, e, 2 or your own, enter the exponent, and the result comes back with the inverse relationship spelled out and the round trip checked. Bases of zero or less, and base 1, are rejected with the reason.Bernoulli ODE SolverSolves y′ + P(x)y = Q(x)yⁿ by showing the substitution v = y^(1−n) that turns it into a linear equation, then integrating. The two degenerate cases are handled explicitly: n = 0 is already linear, n = 1 is separable.Convolution CalculatorLinear and circular discrete convolution with the sliding-and-summing steps shown term by term, plus a symbolic continuous convolution solved exactly through the Laplace convolution theorem.

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 is the one to open when you need it to 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 →
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 Flat Pattern of a Cone Is a Sector, Not a CircleRoll a 50 mm cone 80 mm tall and its pattern is a 190.8° slice of a 94 mm circle. Flatten a nearly cylindrical lampshade and the pattern radius runs to 839 mm — the shallower the taper, the further away the apex.ExplainerA 1 % Drop Rate Does Not Mean One Hundred RunsAt 1 %, a hundred attempts give you 63.4 % — not certainty. Ninety per cent takes 230 attempts and ninety-nine takes 459, and more than a third of players are still empty-handed at a hundred.ExplainerFibonacci and the Golden RatioThe Fibonacci sequence adds each pair of terms to make the next; the ratio of neighboring terms homes in on the golden ratio φ ≈ 1.618. See how, and where the pattern turns up.ExplainerWhat Is a Z-Score? Standard Deviations Above the MeanA z-score is z = (x − μ) / σ — how many standard deviations a value sits from the mean. Learn how to compute it, read the standard normal curve, and turn it into a percentile.