Taylor's and Maclaurin's series

Taylor's and Maclaurin's Series Taylor's and Maclaurin's series are powerful methods used to approximate functions with infinite accuracy. These series are...

Taylor's and Maclaurin's Series

Taylor's and Maclaurin's series are powerful methods used to approximate functions with infinite accuracy. These series are constructed using a sequence of derivatives and integrals.

Taylor's Series:

Suppose a function f(x) has a derivative at x = a.
The Taylor series for f(x) centered at a is an infinite sum of terms:

$f(x) \approx f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots$

The first few terms of the Taylor series typically suffice to give a good approximation of f(x) near a.

Maclaurin's Series:

Suppose a function f(x) has a second derivative at x = a.
The Maclaurin series for f(x) centered at a is an infinite sum of terms:

$f(x) \approx f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots$

The first few terms of the Maclaurin series typically suffice to give a good approximation of f(x) near a.

Examples:

The Taylor series for f(x) = sin(x) centered at a = 0 is:

$\sin(x) \approx x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$

The Maclaurin series for f(x) = e^x centered at a = 0 is:

$e^x \approx 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$