How to use the Taylor series calculator
A Taylor series turns a curve into a polynomial you can actually add up. Pick a function, a value of x and a number of terms, and the calculator sums the series expanded around x = 0, prints it next to the true value, and tabulates how the error shrinks as each term joins in. Summing five terms of eˣ at x = 1 gives 2.70833…, while e itself is 2.71828…, so the error is about 0.00995.
The point that matters most is that convergence depends on the function. The series for eˣ, sin x and cos x converge for every real x, so more terms always close the gap. The series for ln(1+x) converges only while |x| is below 1, converges painfully slowly at x = 1, and the function is undefined for x of −1 or less. The series for 1÷(1−x) also needs |x| below 1, and at x = 1 the function has no value at all. Outside those windows the calculator explains the problem instead of printing a number.
Each term is built from the previous one by multiplying a ratio, so growing factorials never overflow. Five functions are supported; arbitrary typed expressions and expansions centred anywhere other than x = 0 are not. The true value is taken from the browser built-in functions, which means the approximation is checked against something independent every time.
Frequently asked questions
Inside the radius of convergence, yes. For ln(1+x) or 1÷(1−x), which need |x| below 1, extra terms push the sum further away once you leave that range.
Five: eˣ, sin x, cos x, ln(1+x) and 1÷(1−x). Typed expressions and composite functions are not supported.
No. This calculator only expands around x = 0, and expansions centred at another point are not supported.