Taylor Series Calculator

About Taylor Series Calculator

A Taylor series approximates a function as an infinite sum of terms. More terms give better accuracy.

How to use this calculator

1. Enter the function you want to approximate, such as \(\sin x\) or \(e^x\).
2. Choose the center point \(a\), where the series is built, for example \(a=0\).
3. Set the number of terms \(N\), which controls how many derivatives are used.
4. Review the approximation, then compare it with the original function near \(a\).

The formula explained

The Taylor series formula gives a polynomial approximation of \(f(x)\) near the point \(a\). It uses the function value and derivatives at \(a\) to estimate values of the function at nearby \(x\)-values.

• f(x) = the function being approximated
• a = the center point where the expansion is taken
• N = the highest derivative term included
• \(f^{(n)}(a)\) = the nth derivative of the function evaluated at \(a\)
• n! = the factorial of \(n\), used to scale each term
• x = the input value where you want to estimate the function

Step by step method

1. Find the function's derivatives up to the order you need.
2. Evaluate each derivative at the center point \(a\).
3. Plug those values into \(\sum_{n=0}^{N} \frac{f^{(n)}(a)}{n!}(x-a)^n\).
4. Simplify the polynomial and use it to estimate the function near \(a\).

Worked example

Problem. Use a Taylor series centered at \(a=0\) to approximate \(e^{0.2}\) with terms through \(N=3\).

1. For \(f(x)=e^x\), every derivative is still \(e^x\), so at \(a=0\), each derivative value is \(1\).
2. Substitute into the formula: \(e^x \approx 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!}\).
3. At \(x=0.2\), this gives \(1 + 0.2 + \frac{0.2^2}{2} + \frac{0.2^3}{6} = 1.221333\ldots\).

Tips and common mistakes

• Taylor series work best near the center point \(a\), so accuracy usually drops as \(x\) moves away from \(a\).
• Be careful with factorials and derivative values, because a small mistake changes every term in the approximation.

Frequently asked questions