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Taylor Series Calculator

Taylor series calculator and Taylor expansion calculator: polynomial of order n about any center, value at x, steps and the Lagrange error bound.

Created by Muhammad Asif Zaheer
Reviewed by CalculatorPlanet Editorial Team

Published

Last updated

The formula
f(x) = ex, order n = 4, center a = 0, evaluate at x = 1 P4(x) = f(a) + f'(a)(x − a) + f''(a)/2! (x − a)² + ... + f(4)(a)/4! (x − a)⁴ Each coefficient is c_k = f(k)(a) / k!, the k-th derivative at the center divided by k factorial.
Derivatives at the center (k = 0 to 4)
f(a) = 1 f(1)(a) = 1 f(2)(a) = 1 f(3)(a) = 1 f(4)(a) = 1
Coefficients c_k = f(k)(a) / k!
c0 = 1 c1 = 1 c2 = 1/2 c3 = 1/6 c4 = 1/24
Maclaurin polynomial (center 0)
P4(x) = 1 + x + x²/2 + x³/6 + x⁴/24
Evaluate at x = 1 (x - a = 1)
k = 0: term 1, running total 1 k = 1: term 1, running total 2 k = 2: term 0.5, running total 2.5 k = 3: term 0.1666666667, running total 2.666666667 k = 4: term 0.04166666667, running total 2.708333333 P4(1) = 2.708333333
Compare with the true value
f(1) = 2.718281828 Actual error |f(x) - P4(x)| = 0.009948495126
Lagrange error bound
|R4(x)| ≤ M x |x - a|⁵ / 5!, where M is any number with |f(5)(t)| ≤ M for every t between a and x. M = 2.718281828 |x - a|⁵ / 5! = 0.008333333333 Error bound = 0.02265234857 The actual error is within the bound, as the theorem promises. The bound is a ceiling, so the real error is often smaller.
Does the full series converge here?
This series converges for every real x (infinite radius of convergence).
Result
P4(1) = 2.708333333 | error bound: 0.02265234857
Taylor polynomial value P_n(x)
2.708333
True value f(x)
2.718282
Actual error |f(x) - P_n(x)|
0.0099484951
Lagrange error bound
0.0226523486
M, bound on |f^(n+1)|
2.7183
x inside the radius? (1 = yes)
1

How the Taylor Series Calculator Formula Works

A Taylor polynomial replaces a complicated function with a simple polynomial that matches it near one point, the center a. It matches the function's value, its slope, its curvature and so on, up to the order n you choose. Each term is the k-th derivative at a, divided by k factorial, times (x − a) raised to the power k.

When the center is 0 the polynomial is called a Maclaurin polynomial. The gap between the polynomial and the true function is the remainder R_n. The Lagrange form says it equals the next derivative, taken at some unknown point c between a and x, divided by (n + 1) factorial, times (x − a) to the power n + 1.

You cannot know c, but you can bound the next derivative by a number M on the interval. That gives a guaranteed ceiling on the error: M times |x - a| to the power n + 1, divided by (n + 1) factorial. The calculator computes M for each supported function.

P_n(x) = sum for k = 0..n of fk(a)/k! (x − a)k; |R_n(x)| ≤ M |x − a|n+1 / (n+1)!
What goes in, what comes out
  • n4Order of the polynomial, the highest power of (x - a).
  • a0Center of the expansion. 0 gives a Maclaurin polynomial.
  • x1The point where the polynomial is evaluated.
  • MAn upper bound for |f^(n+1)| between a and x.
  • RRadius of convergence of the full series about a.
Taylor polynomial value P_n(x)
2.708333
Values shown are from the worked example below
n
n/a
Order of the polynomial, the highest power of (x - a).
a
n/a
Center of the expansion. 0 gives a Maclaurin polynomial.
x
n/a
The point where the polynomial is evaluated.
M
n/a
An upper bound for |f^(n+1)| between a and x.
R
n/a
Radius of convergence of the full series about a.

Using the Taylor Series Calculator

  1. Enter function f(x)

    Choose from e^x, sin x (radians), cos x (radians), ln(1 + x), 1/(1 - x), arctan x, (1 + x)^p (binomial series).

  2. Enter order n (highest power, 0 to 20)

    Any value between 0 and 20.

  3. Enter center a (0 gives a maclaurin polynomial)

    Type the figure you already have, then press Calculate.

  4. Enter evaluate at x

    Type the figure you already have, then press Calculate.

  5. Enter exponent p

    Type the figure you already have, then press Calculate.

  6. Read the result

    Press Calculate to update the result. Until you do, the last result is dimmed and marked as out of date, so you always know which numbers belong to the values in the boxes.

A Real Worked Example

Approximate e at x = 1 with a Maclaurin polynomial of order 4. Every derivative of ex at 0 is 1, so the polynomial is 1 + x + x²/2 + x³/6 + x⁴/24. At x = 1 it adds up to 1 + 1 + 0.5 + 0.166667 + 0.041667 = 2.708333.

The true value of e is 2.718282, so the actual error is 0.009948. For the Lagrange bound, the fifth derivative of ex is et, at most e = 2.7183 on [0, 1]. The bound is e / 5! = 0.02265, and the actual error sits safely below it.

1
fn
exp
2
n
4
3
a
0
4
x
1
5
p
1/2
This calculator returns2.708333approximation2.718282exactValue0.0099484951actualError0.0226523486errorBound2.7183derivativeBound1insideRadius

Sine with exact fractions

For sin x with order 5 about 0, the polynomial is x - x³/6 + x⁵/120. At x = 1 that is 1 − 0.166667 + 0.008333 = 0.841667.

The true sin 1 is 0.841471, so the error is 0.0001957. All derivatives of sine are at most 1 in size, so M = 1 and the bound is 1 / 6! = 0.001389. Sine's even terms vanish about 0, which is why order 5 has only three nonzero terms.

fn: sinn: 5a: 0x: 1p: 1/20.841667approximation0.841471exactValue0.0001956819actualError0.0013888889errorBound1derivativeBound1insideRadius

A center other than 0

Expand cos x about a = π/4 with order 3 and evaluate at x = π/3. Then x − a = π/12 = 0.261799. The derivatives at π/4 follow the cosine cycle, giving coefficients 0.707107, -0.707107, -0.353553 and 0.117851.

The polynomial value is 0.499869 against a true cos(π/3) = 0.500000. The actual error is 0.0001309 and the bound, with M = 1, is 0.0001957. Choosing a center near x gives high accuracy with few terms.

fn: cosn: 3a: pi/4x: pi/3p: 1/20.499869approximation0.5exactValue0.0001308531actualError0.0001957326errorBound1derivativeBound1insideRadius

ln(1 + x) near the edge of its radius

For ln(1 + x) the Maclaurin series is x - x²/2 + x³/3 - x⁴/4 + x⁵/5. At x = 0.5 the order 5 polynomial gives 0.407292, and the true ln 1.5 is 0.405465.

The actual error is 0.001827. For the bound, the sixth derivative is 5!/(1 + t)⁶, at most 120 on [0, 0.5], so the bound is 120 × 0.5⁶ / 720 = 0.002604. The radius of convergence is 1, so x = 0.5 is inside.

fn: ln1pn: 5a: 0x: 0.5p: 1/20.407292approximation0.405465exactValue0.0018265586actualError0.0026041667errorBound120derivativeBound1radius1insideRadius

Arctan

The arctan series is x - x³/3 + x⁵/5 - .... With order 5 at x = 0.5 the polynomial is 0.5 − 0.041667 + 0.006250 = 0.464583, against a true 0.463648.

The actual error is 0.0009357. The calculator bounds the sixth derivative by 5! = 120, which gives an error bound of 0.002604. This bound is valid but looser than the real error, a normal feature of Lagrange bounds.

fn: atann: 5a: 0x: 0.5p: 1/20.464583approximation0.463648exactValue0.0009357243actualError0.0026041667errorBound120derivativeBound1radius1insideRadius

A binomial series

Approximate the square root of 1.2 as (1 + x)1/2 at x = 0.2. The binomial coefficients for p = 1/2 are 1, 1/2, -1/8 and 1/16, so the order 3 polynomial is 1 + x/2 - x²/8 + x³/16 = 1.095500.

The true square root is 1.095445, so the error is 5.488e-05. The fourth derivative has coefficient |(1/2)(-1/2)(-3/2)(-5/2)| = 0.9375, giving a bound of 6.25e-05.

fn: binomn: 3a: 0x: 0.2p: 1/21.0955approximation1.095445exactValue0.000054885actualError0.0000625errorBound0.9375derivativeBound1radius1insideRadius

What Else to Know

What a Taylor series calculator does

A Taylor series turns a function into an endless sum of powers of (x − a), built from the function's derivatives at the center a. Cut the sum off after the term of order n and you have the Taylor polynomial. Near a, the polynomial is very close to the function, and it is easy to compute because it needs only addition and multiplication.

Diagram of the Taylor polynomial formula, the coefficient f^(k)(a)/k! and the Lagrange remainder bound

This Taylor series calculator, also a Taylor expansion calculator, builds that polynomial for a function you choose from a list, evaluates it at your x, and compares it with the true value. It then gives the Lagrange error bound, which is the guaranteed ceiling on how far off the polynomial can be.

The function comes from a drop-down list on purpose. Reading an arbitrary typed function exactly and safely is not something a small page can promise, and a wrong guess would give wrong derivatives. The seven supported functions have known closed-form derivatives, so every coefficient and bound is exact.

The Taylor and Maclaurin formula, and how to find a polynomial by hand

The Taylor polynomial of order n about a is P_n(x) = f(a) + f'(a)(x − a) + f''(a)/2! (x − a)² + ... + fn(a)/n! (x − a)n. OpenStax Calculus Volume 2 states it exactly this way, as the n-th partial sum of the Taylor series. When a = 0 it is called a Maclaurin polynomial.

Worked example flow for e^x, order 4 at x equals 1: polynomial 2.708333, true e 2.718282, bound 0.02265

The quantity fk(a)/k! is the coefficient c_k. For ex about 0, every c_k is 1/k!. For sin x the derivatives cycle through sin, cos, minus sin and minus cos, which is why only odd powers appear about 0. The first table lists the Maclaurin series for each supported function.

Graph of e^x with Maclaurin polynomials of order 0 to 4 hugging the curve near zero

A useful habit is to count terms by order. Order n has n + 1 terms, from k = 0 up to k = n. Order 0 is just the constant f(a), order 1 is the tangent line, and order 2 adds curvature.

Graph of sin x with Maclaurin polynomials of order 1, 3, 5 and 9, wider agreement at higher orders

Pick the function, the center and the order. Take derivatives up to order n, evaluate each at a, divide by k!, and attach (x − a)k. For ex about 0 with n = 4, every derivative at 0 is 1, so you get 1 + x + x²/2 + x³/6 + x⁴/24.

Then evaluate at your x. At x = 1 the five terms add up to 2.708333, close to e = 2.718282. Off-center is the same recipe with more arithmetic. For cos x about π/4 the coefficients are 0.7071, -0.7071, -0.3536 and 0.1179, from cos(π/4) and sin(π/4) both equal to about 0.7071.

The steps view of the calculator prints exactly this chain: derivative values, coefficients, the polynomial, and the running total term by term. The table of the sine and cosine derivative cycle shows the pattern that makes these two easy.

The Lagrange remainder and the error bound

The remainder is R_n(x) = f(x) - P_n(x). Taylor's theorem with remainder, as stated in OpenStax Calculus Volume 2, says that if f can be differentiated n + 1 times on an interval containing a and x, then R_n(x) = fn+1(c)/(n+1)! (x − a)n+1 for some c between a and x. If |fn+1| ≤ M on that interval, then |R_n(x)| ≤ M |x - a|n+1 / (n+1)!.

The theorem does not tell you c, so you replace the unknown derivative by its largest possible size M. The calculator finds M from a closed form for each function, as the table on this page shows. For ex at x = 1, M is e, and for sine and cosine M is 1.

Remember that the bound is a ceiling. For ex with n = 4 at x = 1 the bound is 0.02265 while the actual error is 0.009948. The next table shows both numbers shrinking as the order rises. Use the bound to choose n: raise it until the bound drops below the accuracy you need.

Bar chart of digits of accuracy for e^x at x equals 1, actual error against Lagrange bound, orders 0 to 8

Choosing the center and the order

A center close to x needs far fewer terms. Expanding sine about 0 and evaluating at x = π/4 ≈ 0.7854 with order 3 leaves an error of 0.00245. Expanding about π/6 and using the same order leaves 0.000107. The center table above shows the gap for several orders.

Graph of sin x with an order 3 polynomial centered at 0 and another centered at pi over 6

The order matters as well. Each extra order multiplies the bound by about |x - a| / (n + 2), so for |x - a| below 1 the error drops quickly. For |x - a| well above 1, the first few terms grow before they shrink, and sine at x = 3 needs order 11 for an error below 0.001.

Graph of actual error and Lagrange bound for e^x order 4 as x moves away from the center

You can type centers and x values as multiples of pi, like pi/4 or 2pi/3. That makes it easy to expand trigonometric functions around the usual angles. Keep angles in radians: a calculator that uses degrees for sine would break the series.

Radius of convergence and when the series fails

The infinite series converges to f(x) only within a certain distance R of the center. For ex, sin x and cos x, R is infinite. For ln(1 + x) and 1/(1 − x) about 0, R is 1, because each function has a trouble spot at distance 1 (x = -1 and x = 1). For arctan x the limit comes from the complex points i and -i, so R is also 1 about 0. About a center a, the radii shift: ln(1 + x) has R = 1 + a, and arctan x has R = √(1 + a²).

Inside the radius, more terms give a better answer. Outside it, more terms give a worse answer. For ln(1 + x) at x = 1.5, order 5 gives 1.753 and order 20 gives -97, while the true value is 0.9163. The calculator flags x as outside the radius so you know to move the center.

Graph of ln(1+x) with order 5 and order 20 polynomials that diverge beyond radius 1

Endpoints need separate tests that this calculator does not do. The ln(1 + x) table shows the inside and outside cases side by side.

How to use this Taylor series calculator

Choose the function, the order n, the center a and the point x. For the binomial series (1 + x)p, enter the exponent p as a whole number, a decimal or a fraction like 1/2 or -1/3. Use pi for pi, as in pi/4 or 2pi/3.

The results show the polynomial value P_n(x), the true value f(x), the actual error, the Lagrange error bound, the derivative bound M, the radius of convergence and whether x lies inside it. The solution view prints the polynomial, the derivative list and each step in plain text.

If the function is not defined between your center and x, the calculator tells you instead of giving a number. For ln(1 + x) the center and x must both exceed -1, and for 1/(1 − x) they must be on the same side of 1. This calculator handles one variable. For anything outside the list, use it as a worked pattern and take derivatives by hand.

Limits of the Taylor series calculator: seven functions, orders to 20, radians, bound caveats

Common mistakes with Taylor polynomials

Using degrees is the classic one: the series for sine and cosine are in radians. The second is forgetting the shift. If the center is not 0, the powers are of (x − a), not x. The third is mixing up order and number of terms, which differ by one.

Another frequent slip is using the bound as if it were the error. It guarantees the error is no larger, nothing more. A very large bound only tells you that the proof was loose, not that the polynomial is bad. Compare it with the actual error the calculator prints next to it.

Finally, in the Lagrange formula the derivative is the (n + 1)th, and the denominator is (n + 1)!, not n!. The mistakes table lists these with fixes. If your own answer differs from the calculator's, compare coefficient by coefficient using the steps.

Related calculus and algebra tools

Taylor polynomials are closely tied to the limits and series work in a first or second calculus course. If you are preparing for an exam where series show up, the AP Calculus BC score calculator estimates your score from section results, and the AP Calculus AB score calculator does the same for AB.

For polynomial work, the polynomial long division calculator and the synthetic division calculator show the steps of dividing the polynomials you get from a Taylor expansion. The linear interpolation calculator is the order 1 case in spirit: a straight line through two known points, with its own error formula.

The definitions and the remainder theorem used here follow OpenStax Calculus Volume 2, section 6.3, and the Lamar University notes on Taylor series, both listed in the references.

Reference Tables

Maclaurin series (center 0) for the functions this calculator supports
FunctionSeriesConverges for
ex1 + x + x²/2! + x³/3! + ...all x
sin xx - x³/3! + x⁵/5! - ...all x
cos x1 - x²/2! + x⁴/4! - ...all x
ln(1 + x)x - x²/2 + x³/3 - x⁴/4 + ...-1 < x ≤ 1
1/(1 − x)1 + x + x² + x³ + ...-1 < x < 1
arctan xx - x³/3 + x⁵/5 - ...-1 ≤ x ≤ 1
(1 + x)p1 + p x + p(p-1)/2! x² + ...-1 < x < 1 (all x if p is a whole number 0 or more)
e^x at x = 1: the polynomial, the true error and the Lagrange bound for each order
Order nP_n(1)Actual errorError bound
01.0000001.7182.718
12.0000000.71831.359
22.5000000.21830.453
32.6666670.051620.1133
42.7083330.0099480.02265
52.7166670.0016150.003775
62.7180560.00022630.0005393
72.7182542.786e-056.742e-05
82.7182793.059e-067.491e-06
sin x: the actual error of the Maclaurin polynomial at two values of x
Order nError at x = 1Error at x = 3Bound at x = 3
10.15852.8594.5
30.0081381.6413.375
50.00019570.38391.012
72.731e-060.050050.1627
92.489e-080.0041920.01627
111.598e-100.00024540.001109
How the calculator picks M, the bound on the (n + 1)th derivative between a and x
Function(n + 1)th derivativeM used
exetex or ea, whichever is larger
sin x, cos x±sin t or ±cos t1
ln(1 + x)±n! / (1 + t)n+1n! / (1 + m)n+1, m = smaller of a and x
1/(1 − x)(n+1)! / (1 − t)n+2(n+1)! / dn+2, d = distance from 1 to the nearest end
arctan xat most n! / (1 + t²)^((n+1)/2)n! / (1 + t0²)^((n+1)/2), t0 = point nearest 0
(1 + x)pp(p-1)...(p-n) (1 + t)p−n−1the product times the largest (1 + t) power
Results for the six worked examples
ExampleSetupP_n(x)Actual errorBound
1ex, n = 4, a = 0, x = 12.7083330.0099480.02265
2sin x, n = 5, a = 0, x = 10.8416670.00019570.001389
3cos x, n = 3, a = π/4, x = π/30.4998690.00013090.0001957
4ln(1 + x), n = 5, a = 0, x = 0.50.4072920.0018270.002604
5arctan x, n = 5, a = 0, x = 0.50.4645830.00093570.002604
6(1 + x)1/2, n = 3, a = 0, x = 0.21.0955005.488e-056.25e-05
Why the center matters: sin x near x = π/4 ≈ 0.7854 with two different centers
Order nError, center 0Error, center π/6
10.078290.01962
20.078290.002483
30.0024540.0001065
40.0024548.638e-06
53.626e-052.378e-07
ln(1 + x) with a Maclaurin polynomial: it settles inside the radius and drifts outside
xP_5(x)P_20(x)True valueInside radius 1?
0.250.223180.223140.22314yes
0.50.407290.405470.40547yes
0.90.692070.639050.64185yes
1.51.75313-96.828580.91629no
25.06667-34359.708091.09861no
How the error grows with distance from the center: e^x, order 4, center 0
xTrue errorLagrange bound
0.18.474e-089.21e-08
0.50.00028380.0004294
10.0099480.02265
20.38911.97
33.71140.67
The derivative cycle for sin x and cos x at the center a
kf^(k)(a) for sin xf^(k)(a) for cos x
0sin acos a
1cos a-sin a
2-sin a-cos a
3-cos asin a
4sin a (repeats)cos a (repeats)
Special cases of the binomial series (1 + x)^p
pWhat it isFirst terms
-11/(1 + x)1 − x + x² - x³
1/2√(1 + x)1 + x/2 - x²/8 + x³/16
-1/21/√(1 + x)1 - x/2 + 3x²/8 − 5x³/16
2(1 + x)²1 + 2x + x² (stops: exact)
3(1 + x)³1 + 3x + 3x² + x³ (stops: exact)
Common mistakes with Taylor polynomials
MistakeWhat goes wrongFix
Using degrees for sin and cosThe series needs radiansConvert: 30 degrees is π/6
Forgetting (x − a) when a is not 0Terms are powers of x − a, not xCompute t = x - a first
Counting the number of terms as the orderOrder n has n + 1 terms, k = 0 to nOrder = highest power
Trusting the polynomial outside the radiusMore terms make it worseCheck |x - a| against R
Reading the bound as the errorThe bound is a ceilingThe real error is usually smaller
Mixing up n and n + 1 in the remainderBound uses the (n + 1)th derivativeR_n uses fn+1 and (n+1)!

Frequently Asked Questions About the Taylor Series Calculator

What is a Taylor series calculator?

It builds a polynomial that approximates a function near a chosen center a, using the function's derivatives there. You pick the function, the order n, the center a and a point x, and it returns the polynomial, its value at x and an error estimate. For ex with n = 4 and x = 1 the value is 2.708333 against a true 2.718282.

How does the Taylor expansion calculator work?

A Taylor expansion calculator, which is what this page is, uses the formula P_n(x) = sum of fk(a)/k! x (x − a)k for k = 0 to n. Each coefficient comes from a closed form for the function you chose, so no numerical differentiation is involved. Then it evaluates the polynomial at x, compares with the true value, and computes the Lagrange bound. The steps view shows each stage.

Why can't I type my own function?

Parsing arbitrary typed functions safely and exactly is hard, and a wrong parse would give a wrong answer without warning. So this calculator offers a fixed list: ex, sin x, cos x, ln(1 + x), 1/(1 − x), arctan x and (1 + x)p. Each has an exact formula for its derivatives and a proven error bound. Many other functions can be rewritten using these, for example ln x is ln(1 + u) with u = x − 1.

What is the difference between Taylor and Maclaurin series?

A Maclaurin series is a Taylor series with the center a = 0. Any Taylor series about another center, like π/4, is just called a Taylor series. Set a to 0 and this calculator writes the exact Maclaurin polynomial with fractions, for example sin x gives x - x³/6 + x⁵/120. With another center it shows decimal coefficients and powers of (x − a).

What is a in the Taylor series?

The letter a is the center, the point where the derivatives are taken and where the approximation is exact. The polynomial is accurate near a and gets worse as x moves away. For cos x with center π/4 and x = π/3, the order 3 value is 0.499869 and the true value is 0.500000, off by only 0.000131.

What is the Taylor polynomial formula?

P_n(x) = f(a) + f'(a)(x − a) + f''(a)/2! (x − a)² + ... + fn(a)/n! (x − a)n. The number n is the order, the highest power of (x − a). There are n + 1 terms, counting the constant. The remainder R_n(x) = f(x) - P_n(x) is the error, and the Lagrange form says it equals fn+1(c)/(n+1)! (x − a)n+1 for some c between a and x.

How do I find the error of a Taylor polynomial?

Compute the true value minus the polynomial value, or bound it with the Lagrange remainder: |R_n(x)| ≤ M |x - a|n+1 / (n+1)!, where M bounds |fn+1| between a and x. For ex at x = 1 with n = 4, M = e, so the bound is e/120 = 0.02265. The actual error is 0.009948.

Is the error bound the same as the actual error?

No. The bound is a guaranteed ceiling, not a prediction. The actual error is usually smaller. For 1/(1 − x) at x = 0.5 with n = 3 the actual error is 0.125 but the bound is 2, because M must cover the worst case anywhere on the interval. A bound that is too large is still correct, just not tight.

How many terms do I need for a given accuracy?

Raise n until the Lagrange bound drops below your tolerance. For ex at x = 1 and a tolerance of 0.000001, the bound is 7.49e-06 at n = 8 and 7.49e-07 at n = 9, so n = 9 is enough. For sin x at x = 1, n = 9 is enough.

What is the radius of convergence?

It is the distance from the center within which the infinite Taylor series converges to the function. ex, sin x and cos x converge everywhere. ln(1 + x) and 1/(1 − x) about 0 have radius 1, set by the trouble spots at x = -1 and x = 1. arctan x about 0 also has radius 1, because of the complex points ±i. The calculator reports R and whether your x is inside.

Does the calculator show the interval of convergence?

It shows the radius R about your center and tells you whether |x - a| is below it. It does not test the endpoints, where convergence can go either way. For example ln(1 + x) about 0 converges at x = 1 but not at x = -1. For x = 0.9 with n = 20 the polynomial gives 0.63905 against a true 0.64185.

What happens if x is outside the radius?

The polynomial can still be evaluated, but adding terms makes it worse, not better. For ln(1 + x) at x = 1.5, P_5 is 1.7531 and P_20 is -96.8, while the true value is 0.9163. The calculator flags this case. Move the center closer to x to fix it.

Can I use degrees for sin and cos?

No. Taylor series for sin x and cos x need radians. Convert first: 30 degrees is π/6, 45 degrees is π/4. You can type pi/6, pi/4 or 2pi/3 directly in the center and x fields. For sin x with center π/6, n = 4 and x = π/4 (45 degrees), the polynomial gives 0.707098 against a true 0.707107.

What are some Taylor series examples?

Classic ones are ex = 1 + x + x²/2 + ..., sin x = x - x³/6 + ..., 1/(1 − x) = 1 + x + x² + ... and √(1 + x) = 1 + x/2 - x²/8 + .... With √(1 + x), n = 3 and x = 0.2 the value is 1.095500, a true 1.095445. The six worked examples on this page cover each function type.

Does it handle multivariable Taylor series?

No. This calculator handles functions of one variable. A multivariable Taylor polynomial needs partial derivatives and a different remainder formula, and the input would be a full expression in several variables, which this tool does not accept. For one variable along a path you can often substitute, but check the method in a calculus text for anything beyond single-variable work.

Is there a Taylor series calculator app?

You do not need an app. This page works in any phone or desktop browser, with no download or sign-up, and the layout is built for small screens. The solution view prints in plain text, so you can copy the polynomial into notes or a document. Type pi/4 and similar multiples of pi directly.

How do I write ln x as a Taylor series?

Shift the variable. With u = x − 1, ln x = ln(1 + u), so choose the ln(1 + x) option, keep the center at 0 and enter u = x − 1 as the point. To approximate ln 1.5, enter x = 0.5. With n = 5 the polynomial gives 0.407292 against a true 0.405465. This works for u between -1 and 1, so x from 0 to 2.

Why is the polynomial exact for (1 + x)^3?

When p is a whole number 0 or more, the binomial series stops after p + 1 terms, so the polynomial is the function. Orders beyond p add zero terms, the derivative bound is 0, and the error is exactly 0 for every x, even below -1. For example (1 + x)3 at x = -2 with n = 3 gives -1 both ways.

How accurate is the calculator?

Coefficients use closed forms in double precision, and the Maclaurin polynomials are written with exact fractions. Results were checked against independent calculations for every function, including off-center cases. Very large x or order values can overflow and the calculator then says so. The bound uses standard textbook derivative estimates, so it is rigorous but not always tight.

Does this calculator show the steps?

Yes. The solution view lists the formula, the derivative values at the center, the coefficients, the polynomial, the running total at your x, the comparison with the true value, the Lagrange bound and the convergence check. You can switch to a result-only view with the button. Use the steps to check your own homework, line by line.

What are the limits of this calculator?

It supports seven functions, orders 0 to 20, and a single variable. It does not accept typed functions, and it reports only the radius of convergence, not endpoint behavior. The error bound assumes the function is defined and smooth between a and x; the calculator stops with a message when it is not, such as 1/(1 − x) with the point past x = 1.

References

  1. [1] OpenStax. “Calculus Volume 2, Section 6.3 Taylor and Maclaurin Series.” OpenStax, Rice University. Accessed 2026-10-07.
  2. [2] Paul Dawkins. “Calculus II: Taylor Series.” Paul's Online Math Notes, Lamar University. Accessed 2026-10-07.
  3. [3] Wikipedia contributors. “Taylor's theorem.” Wikipedia. Accessed 2026-10-07.
  4. [4] Wikipedia contributors. “Taylor series.” Wikipedia. Accessed 2026-10-07.
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