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Calculus Solver

Taylor Series Calculator

Build the Taylor polynomial of a function about any center, or a Maclaurin polynomial about 0. The calculator lists each derivative, its exact value at the center and the resulting coefficient, so you can follow the construction term by term.

Your expression will be shown here as it will be read.

A center of 0 gives the Maclaurin polynomial.

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Try an example

What is a Taylor polynomial?

Several approximating curves hugging an original curve closely near one central point

A Taylor polynomial approximates a smooth function near a point a using the function's derivatives at that single point. The degree-n polynomial matches f and its first n derivatives at a, so it hugs the graph closely near the center and usually drifts away further out.

Letting n grow without bound gives the Taylor series. When the center is a = 0, the polynomial and series are called Maclaurin polynomials and series. The calculator computes the polynomial up to the order you choose, which is what most homework and approximation problems need.

Tn(x)=∑k=0nf(k)(a)k! (x−a)kT_{n}(x) = \sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}\,(x - a)^{k}

Series worth knowing by heart

These Maclaurin series appear constantly. The calculator reproduces them, and you can check each against its output.

  • Exponential

    ex=1+x+x22!+x33!+⋯e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \cdots

    Converges for every real x.

  • Sine

    sin⁡x=x−x33!+x55!−⋯\sin x = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \cdots

    Only odd powers, because sine is an odd function.

  • Cosine

    cos⁡x=1−x22!+x44!−⋯\cos x = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \cdots

    Only even powers, because cosine is even.

  • Geometric series

    11−x=1+x+x2+x3+⋯\frac{1}{1 - x} = 1 + x + x^{2} + x^{3} + \cdots

    Converges only for |x| < 1.

  • Natural log

    ln⁡(1+x)=x−x22+x33−⋯\ln(1 + x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \cdots

    Converges for −1 < x ≤ 1.

  • Lagrange remainder

    Rn(x)=f(n+1)(c)(n+1)!(x−a)n+1R_{n}(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x - a)^{n+1}

    For some c between a and x; use it to bound the approximation error.

How this calculator works

The calculator differentiates your function repeatedly using the same symbolic engine as the derivative calculator. It simplifies each derivative before taking the next, so expressions stay manageable.

Each derivative is evaluated at the center exactly: sin(π) = 0, e⁰ = 1, √4 = 2, with no rounding. Each value is then divided by k! to get the coefficient. Terms with a zero coefficient are skipped, and the rest are combined in increasing powers of (x − a).

If the function or one of its derivatives is undefined at the center (for example ln(x) at 0), there is no Taylor polynomial there, and the calculator says so instead of producing nonsense.

For the full details, including how answers are verified, see how the solver works.

Worked examples

Work through these by hand first, then enter them above to compare each step.

Example 1

Maclaurin polynomial of eˣ

T4(x) for ex about 0T_{4}(x)\ \text{for}\ e^{x}\ \text{about}\ 0

  1. Every derivative of eˣ is eˣ, and e⁰ = 1, so each coefficient is 1/k!.

    10!+11!x+12!x2+13!x3+14!x4\frac{1}{0!} + \frac{1}{1!}x + \frac{1}{2!}x^{2} + \frac{1}{3!}x^{3} + \frac{1}{4!}x^{4}

Answer

1+x+x22+x36+x4241 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + \frac{x^{4}}{24}

Example 2

Cosine to order 6

T6(x) for cos⁡x about 0T_{6}(x)\ \text{for}\ \cos x\ \text{about}\ 0

  1. The derivatives cycle cos, −sin, −cos, sin. At 0 they give 1, 0, −1, 0, 1, 0, −1.

    1−x22!+x44!−x66!1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!}

Answer

1−x22+x424−x67201 - \frac{x^{2}}{2} + \frac{x^{4}}{24} - \frac{x^{6}}{720}

Example 3

ln x about a = 1

T4(x) for ln⁡x about 1T_{4}(x)\ \text{for}\ \ln x\ \text{about}\ 1

  1. The derivatives are 1/x, −1/x², 2/x³, −6/x⁴. At x = 1 they equal 1, −1, 2, −6, and ln 1 = 0.

    11!(x−1)−12!(x−1)2+23!(x−1)3−64!(x−1)4\frac{1}{1!}(x-1) - \frac{1}{2!}(x-1)^{2} + \frac{2}{3!}(x-1)^{3} - \frac{6}{4!}(x-1)^{4}

Answer

(x−1)−(x−1)22+(x−1)33−(x−1)44(x-1) - \frac{(x-1)^{2}}{2} + \frac{(x-1)^{3}}{3} - \frac{(x-1)^{4}}{4}

Example 4

Approximating √4.1

T2(x) for x about 4T_{2}(x)\ \text{for}\ \sqrt{x}\ \text{about}\ 4

  1. f(4) = 2, f′(x) = 1/(2√x) gives 1/4, and f″(x) = −1/(4x^{3/2}) gives −1/32. Divide the last by 2! to get −1/64.

    T2(x)=2+x−44−(x−4)264T_{2}(x) = 2 + \frac{x - 4}{4} - \frac{(x - 4)^{2}}{64}

  2. At x = 4.1: 2 + 0.025 − 0.00015625 = 2.02484375. The true value is 2.0248457…, so the error is about 2 × 10⁻⁶.

    4.1≈2.02484375\sqrt{4.1} \approx 2.02484375

Answer

2+x−44−(x−4)2642 + \frac{x - 4}{4} - \frac{(x - 4)^{2}}{64}

Common mistakes

  • Forgetting the factorial

    The coefficient is f⁽ᵏ⁾(a)/k!, not f⁽ᵏ⁾(a). Missing the k! is the single most common error.

  • Using x instead of (x − a)

    Away from a = 0, every term involves powers of (x − a). Expanding in powers of x mixes up the coefficients.

  • Confusing order with number of terms

    “Order 6” means the highest power is 6. For cos x that is only four nonzero terms, because the odd coefficients vanish.

  • Trusting the polynomial far from the center

    The polynomial for 1/(1 − x) is useless at x = 2 because the series diverges for |x| ≥ 1. Stay inside the interval of convergence.

Limitations

  • Polynomials are computed up to order 10. General nth-term formulas and full infinite series are not derived.
  • The radius and interval of convergence are not computed automatically.
  • Functions whose derivatives grow very complicated (high powers of tan x, for instance) may exceed the size limit at higher orders. Lower the order if that happens.
  • Centers must be finite numbers or constant expressions such as pi/2.

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Frequently asked questions

What is the difference between a Taylor series and a Maclaurin series?

A Maclaurin series is simply a Taylor series centered at a = 0. Set the center to 0 in the calculator to get a Maclaurin polynomial.

What does the order of a Taylor polynomial mean?

It is the highest power of (x − a) kept. An order-5 polynomial includes terms up to (x − a)⁵, although some of those coefficients may be zero.

How accurate is a Taylor polynomial?

Very accurate close to the center and less accurate further away. The Lagrange remainder formula bounds the error, and the calculator reports a sample comparison between f and the polynomial near the center.

Why can't I center ln(x) at 0?

ln(x) is undefined at 0, so it has no derivatives there and no Taylor polynomial. Use a = 1, or expand ln(1 + x) about 0 instead.

Next step

Pick one of the worked examples, solve it on paper, then enter it in the calculator and compare your steps with each rule shown. Spotting where your work differs is the fastest way to improve.

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