# Online Taylor Series Calculator

\( \)\( \)\( \)\( \)An online taylor series calculator is presented.

## Definition of Taylor Series

For a function \( f \) whose derivatives are defined in an interval containing \( a \), The Taylor series of function \( f \) at \( x = a \) is given by
\[ \sum_{k=0}^{\infty} \dfrac{f^k(a)}{k!} (x-a)^k = f(a) + f'(a) (x-a) + \dfrac{f''(a)}{2!} (x-a)^2 + ... + \dfrac{f^{(n)}(a)}{n!} (x-a)^n + ... \]

In practice a Taylor polynomial \( P_n(x) \) is obtained by truncating Taylor series to \( n \) terms.

The online calculator presented below, finds the Taylor polynomial at \( x = a \) including \( n \) terms.

## Use of Taylor Series Calculator

1 - Enter and edit function $f(x)$ and click "Enter Function" then check what you have entered.

Note that the five operators used are: + (plus) , - (minus), / (division) , ^ (power) and * (multiplication). (example: f(x) = x*log(x)).(more notes on editing functions are located below)

2 - Click "Calculate Taylor Expansion" to obain the Taylor expansion of the entered function at the values of \( a \) and \( n \) entered.

Note that \( a \) can take integer values only and \( n \) is a positive integer.

Notes: In editing functions, use the following:

1 - The five operators used are: + (plus) , - (minus), / (division) , ^ (power) and * (multiplication). (example: f(x) = x*ln(x+1))

2 - The function square root function is written as (sqrt). (example: sqrt(x^2+1)

Here are some examples of functions that you may copy and paste to practice:

e^x ln(x^2+1) e^(-x^2) 1/(x-2) sin(2*x - 2) sqrt(x^2+1)