Taylor series expansion

This online Taylor Series expansion calculator is designed to facilitate the approximation of functions around a specific point. It allows for the decomposition of a function into a polynomial series, offering a clear view of its local behavior. This tool is especially useful for students, teachers, and professionals looking to analyze complex functions in a simplified manner.

To use this tool, fill in the following fields:

• Main Variable: The function's variable (e.g., x).
• Function: The function to develop (e.g., e^x, sin(x)).
• x0 (Point): The point around which the development is performed (e.g., 0, 1).
• Order (n): The order up to which the development is calculated (e.g., 2, 3).

This guide helps you correctly enter data into the calculator:

Taylor Series expansion is a polynomial approximation of a function around a given point, often used to simplify the analysis of local behaviors of complex functions. It involves expressing a function as a sum of polynomial terms and a remainder, usually as a function of the power of `(x - a)`, where `a` is the development point.

Taylor Series expansion are essential in analysis and calculus for estimating the behavior of functions near a specific point. They are particularly useful for calculating limits, derivatives, and integrals. These approximations also help simplify complex expressions in asymptotic analyses or convergence studies.

To calculate a Taylor Series expansion, it is necessary to know the successive derivatives of the function at the development point. The general formula (or Taylor's formula) for a Taylor Series expansion of order `n` around `a` is given by:

`f(x) ≈ f(a) + f'(a)*(x-a) + (f''(a))/(2!)*(x-a)^2 + ... + f^n(a)/(n!)*(x-a)^n`

Here are some examples of Taylor Series expansion for commonly used functions:

• `e^x ≈ 1 + x + x^2/2! + x^3/3! + ...` around 0.
• `sin(x) ≈ x - x^3/3! + x^5/5! - ...` around 0.
• `cos(x) ≈ 1 - x^2/2! + x^4/4! - x^6/6! + ...` around 0.
• `tan(x) ≈ x + x^3/3 + 2x^5/15 + 17x^7/315 + ...` around 0.
• `ln(1+x) ≈ x - x^2/2 + x^3/3 - ...` around 0.
• `sqrt(1+x) ≈ 1 + x/2 - x^2/8 + ...` around 0.
• `(1+x)^a ≈ 1 + ax + a(a-1)x^2/2! + a(a-1)(a-2)x^3/3! + ...` around 0 (a is a real number).

Here are several examples illustrating the calculation of Taylor Series expansion for different functions using the Taylor method:

• Development of `cos^2(x)` around 0 up to the 2nd order:

  For `f(x) = cos^2(x)`, calculate `f(x_0)`, `f'(x_0)`, and `f''(x_0)` for `x_0 = 0`.

  We have `f(x_0) = cos^2(0) = 1`,

  `f'(x) = -2*cos(x)*sin(x)` so `f'(x_0) = 0`, and

  `f''(x) = -2*cos^2(x) + 2*sin^2(x)` so `f''(x_0) = -2`.

  The Taylor Series expansion is therefore `cos^2(x) ≈ 1 - x^2 + o(x^2)`.

• Development of `1/(1 - x)` around 0 up to the 2nd order:

  With `f(x) = 1/(1 - x)`, determine `f(x_0)`, `f'(x_0)`, and `f''(x_0)` for `x_0 = 0`.

  Here, `f(x_0) = 1/(1 - 0) = 1`,

  `f'(x) = 1/(1 - x)^2` so `f'(x_0) = 1`, and

  `f''(x) = 2/(1 - x)^3` so `f''(x_0) = 2`.

  The Taylor Series expansion is then `1/(1 - x) ≈ 1 + x + x^2 + o(x^2)`.

The Taylor Series expansion calculator is a valuable tool for studying the local behavior of functions. It facilitates the approximation of complex functions and enhances the understanding of fundamental function properties in mathematical analysis.