Taylor series calculator with steps
This Taylor series calculator with steps constructs a finite Taylor polynomial around a chosen center. Maclaurin mode uses the same formula with center 0, so you can use one page as a Maclaurin series calculator with steps. Choose a requested degree from 0 to 10; coefficients above the function’s actual polynomial degree may be zero.
The result is a finite polynomial approximation, not a claim that an infinite series converges to the function. The Derivative Calculator provides separate derivative expressions. Your existing Arithmetic & Geometric Sequence Calculator concerns sequence terms and sums, whereas this page builds a local expansion of an entered function.
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Optional calculus workbooks (paid link) can provide exercises on Taylor coefficients and approximation. The online tool is free, with no purchase needed.
Choose a Taylor center, Maclaurin mode and degree
Enter a supported function and choose x, y or t. In Taylor mode, enter an expansion center such as 1 or 1/2. In Maclaurin mode the center is fixed at 0 and the center input is disabled. Select a requested degree from 0 to 10, then choose Calculate. Leave the optional evaluation point blank to obtain only the polynomial and coefficients.
If you enter an evaluation point, the result compares the polynomial value with the original function value there. This comparison is numerical and is not a proven error bound. Editing any setting clears stale output. Reset restores the initial blank function, Taylor mode, center 0 and degree 5. A graph-paper notebook (paid link) is optional for organizing coefficients by power; the Fraction Calculator can help check factorial divisions.
Expression syntax and angle conventions
Use x, y or t as the single selected variable; numbers; pi and e; parentheses; and +, -, *, / or ^. Supported functions are sin, cos, tan, exp, ln, sqrt, abs, asin, acos and atan. Write function arguments in parentheses. log means the natural logarithm, just like ln. Trigonometric angles and inverse trigonometric values use radians.
Explicit multiplication, such as 2*x, and implicit multiplication, such as 2x or (x+1)(x-1), are accepted. Powers associate to the right: x^2^3 means x^(2^3). A leading minus has lower precedence than a power: -x^2 means -(x^2). Constant fields accept expressions such as 1/3, pi/2 and sqrt(2), but not the selected variable. Use exact rational points when checking a removable singularity.
Taylor polynomial formula and coefficient method
For center a and requested degree n, P_n(x) = Σ[k=0..n] f^(k)(a)*(x-a)^k/k!. The constant coefficient is f(a), the linear coefficient is f′(a), and later coefficients divide the corresponding derivative by its factorial. The Maclaurin polynomial is the case a = 0. The actual resulting degree can be lower than n when high coefficients vanish.
The engine calculates truncated power series through the expression tree. It uses convolution for products, recurrence division for quotients and chain-rule recurrences for supported functions. These give the same Taylor coefficients as repeated differentiation within the supported analytic domain. The steps list each coefficient and the implied derivative value; they do not pretend to be a separate hand differentiation derivation.
Exact rational coefficients are retained where possible. Values involving general trigonometric constants, pi, e or other transcendental evaluations use rounded floating-point coefficients. Common zero-center values such as sin(0), cos(0), exp(0) and ln(1) are recognized exactly. Use the Simplifying Expressions Calculator before entering a complicated expression, while retaining any original domain restrictions.
Maclaurin expansions and Taylor polynomial examples
Exponential: For exp(x), Maclaurin mode and degree 5, the polynomial is 1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120. The denominators come from factorials. At x = 0.1 its value is close to exp(0.1), but a numerical agreement at one point does not prove convergence everywhere.
Sine: For sin(x) around 0 through degree 5, P_5(x) = x - x^3/6 + x^5/120. The even coefficients are zero. For ln(1+x), the first terms through degree 4 are x - x^2/2 + x^3/3 - x^4/4. Optional scientific calculators (paid link) may help check radian-mode function values and factorial arithmetic.
Nonzero center: For x^3 around a = 2 through degree 3, the polynomial is 8 + 12*(x-2) + 6*(x-2)^2 + (x-2)^3. Expanding it returns the original polynomial. The Exponent Calculator can help review individual powers separately.
Reading coefficients and checking an approximation
The main result is written in powers of the selected variable minus the center. The requested degree controls how many coefficients are constructed, including zeros. A leading ≈ means at least one coefficient is numerical rather than an exact rational value. Copy, Download and Print retain the center, degree, coefficients and methodology.
An optional comparison shows the polynomial value, original function value and absolute numerical difference at one entered point. This difference depends on floating-point evaluation and is not a rigorous Taylor remainder estimate. A Taylor polynomial can be evaluated far from its center even where it approximates the function poorly. Optional graphing calculators (paid link) can help compare a polynomial with the original curve; check the viewing window and the model’s supported functions.
The Limit Calculator answers a limiting-value question, and the Integral Calculator handles supported integration tasks. Neither substitutes for proving that an infinite Taylor series converges to its original function.
Analytic domains, supported functions and limitations
The original expression must have a supported real analytic neighborhood around the center. Denominators must be nonzero there. ln arguments and nonconstant sqrt arguments must be positive at the center. Inverse sine and cosine require inner values strictly between -1 and 1. tan must be away from its poles. Noninteger and variable powers require a positive base. abs at an inner zero is conservatively rejected even if a particular composition could simplify to an analytic function.
Do not enter sin(x)/x at center 0 and expect the tool to fill its hole: the original function is undefined there. An explicitly defined analytic extension is a different input. If a supported expression is rejected, this does not prove that no mathematical Taylor polynomial exists; the domain checks are deliberately conservative.
This version supplies finite polynomials through degree 10, not an infinite summation pattern, radius or interval of convergence, rigorous remainder bound, multivariable expansion or graph. Those unimplemented features are not promised by this page. Constant comparisons use the selected variable’s real domain and radian convention.
Expressions are limited to 240 characters, 160 tokens and 24 nested levels. Centers and optional points have magnitude at most 1,000,000. Work, coefficient size, symbolic input power and worker time limits can require a simpler expression or lower degree. Numerical values display about 12 significant digits and should be checked independently when precision matters.
Taylor & Maclaurin Series Calculator FAQs
Sources and methodology
These educational references explain the underlying mathematics. The independent implementation uses the supported methods and restrictions documented above and is not affiliated with the publisher.