Taylor Series Tool for Maths Expansions and Approximations

Taylor Series Calculator Form

Choose a supported function, set the expansion center, pick the order, and evaluate the approximation at any valid point.

Example Data Table

Function	Order n	x	Approximation	Actual Value	Absolute Error
e^x	5	1	2.716667	2.718282	0.001615
sin(x)	7	0.5	0.479426	0.479426	0.000000
ln(1+x)	5	0.3	0.262340	0.262364	0.000024
1/(1-x)	6	0.4	1.663936	1.666667	0.002731

Formula Used

Core Taylor formula
T_n(x) = Σ [ f^(k)(a) / k! ] (x-a)^k, for k = 0 to n

Absolute error
Absolute Error = | f(x) - T_n(x) |

Relative error percentage
Relative Error % = | f(x) - T_n(x) | / | f(x) | × 100

Next-term estimate
Next-Term Estimate ≈ | f⁽ⁿ⁺¹⁾(a) / (n+1)! | |x-a|ⁿ⁺¹

Supported Function Patterns

Function	Derivative Pattern	Domain	Convergence Guide
e^x	Every derivative equals e^x.	All real values are valid.	Infinite radius of convergence.
sin(x)	Derivatives cycle: sin, cos, -sin, -cos.	All real values are valid.	Infinite radius of convergence.
cos(x)	Derivatives cycle: cos, -sin, -cos, sin.	All real values are valid.	Infinite radius of convergence.
sinh(x)	Derivatives alternate between sinh and cosh.	All real values are valid.	Infinite radius of convergence.
cosh(x)	Derivatives alternate between cosh and sinh.	All real values are valid.	Infinite radius of convergence.
ln(1+x)	For n >= 1: (-1)^(n-1)(n-1)! / (1+x)^n.	Requires x > -1 and center a > -1.	Radius equals \|a + 1\|.
1 / (1-x)	n-th derivative: n! / (1-x)^(n+1).	Requires x != 1 and center a != 1.	Radius equals \|1 - a\|.
1 / (1+x)	n-th derivative: (-1)^n n! / (1+x)^(n+1).	Requires x != -1 and center a != -1.	Radius equals \|1 + a\|.

How to Use This Calculator

Select one of the supported functions from the list.

Enter the expansion center a. Use zero for a Maclaurin series.

Enter the evaluation point x where you want the approximation.

Choose the order n. Higher orders usually improve local accuracy.

Set the number of displayed decimals for cleaner reporting.

Press Calculate Series to see the result above the form.

Review the polynomial, term table, errors, and convergence note.

Use the CSV and PDF buttons to download the computed report.

Frequently Asked Questions

1) What does this tool calculate?

It builds a Taylor polynomial for a selected function around a chosen center, evaluates the approximation at x, and reports error measures.

2) What is the difference between Taylor and Maclaurin series?

A Maclaurin series is simply a Taylor series centered at zero. Any nonzero center produces a general Taylor expansion.

3) Why does increasing n often improve accuracy?

Higher-order polynomials include more local derivative information. That usually improves accuracy near the center, though distance from the center still matters.

4) Why can a series still perform poorly?

Some functions have nearby singularities or limited convergence radii. When x is far from the center, the polynomial may converge slowly or fail.

5) What does the next-term estimate mean?

It uses the first omitted term as a practical size check. It is useful for intuition, but it is not a guaranteed rigorous remainder bound.

6) Which functions are supported here?

The tool supports e^x, sin(x), cos(x), sinh(x), cosh(x), ln(1+x), 1/(1-x), and 1/(1+x).

7) Why do domain restrictions matter?

Functions like ln(1+x) and rational forms are not defined everywhere. Invalid centers or evaluation points would make the derivatives or values undefined.

8) Can I export my result?

Yes. After calculating, use the CSV button for spreadsheets or the PDF button for a compact printable summary.