Calculadora de Série de Taylor (e^x)
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
eˣ ≈ Σ_{k=0..N-1} xᵏ/k!About this calculator
The Taylor Series Calculator for e^x approximates the exponential function using the Taylor series expansion centered at x=0 (Maclaurin series). The formula used is e^x ≈ Σ_{k=0}^{N-1} x^k/k!, where N is the number of truncated terms. This means the calculator sums the first N terms of the infinite series, providing an approximation that improves as N increases.
This tool is useful in contexts where direct calculation of e^x is not feasible or when understanding the behavior of polynomial approximation is desired. For example, in numerical analysis, physics, or engineering, the Taylor series is often used to simplify complex calculations. The calculator allows varying x and N, showing how the approximation converges to the true value.
Important considerations: the Taylor series for e^x converges for all real x, but accuracy depends on the number of terms and the value of x. For large |x|, more terms are needed for a good approximation. Additionally, due to floating-point limitations, rounding errors may occur for very large x or N. It is recommended to compare with the exact value of e^x when possible.
Frequently asked questions
How many terms (N) are needed for a good approximation?
It depends on the value of x. For small x (near 0), few terms suffice. For large x, like x=10, dozens of terms are needed for reasonable accuracy.
Does the Taylor series work for negative x values?
Yes, the series converges for all real numbers, including negative x. However, for negative x with large magnitude, more terms are required.
What is the error of the approximation?
The error is given by the Lagrange remainder: |R_N| ≤ e^|x| * |x|^N / N!. It decreases as N increases, and for fixed x, the error tends to zero.
Can I use this calculator for functions other than e^x?
No, this calculator is specific to e^x. For other functions, a different Taylor series would be needed.
What does truncating the series at term N mean?
It means we sum only the first N terms (k=0 to N-1), ignoring the remaining terms. This yields a polynomial approximation of degree N-1.