They provide a foundation for constructing Taylor series for other functions and are frequently encountered.

Taylor series use polynomial approximations to represent functions as infinite series.

The Taylor series approximates the function best near the center 'a'.

The Maclaurin series for $e^x$ contains all the terms from the Maclaurin series of sin(x) and cos(x).

Taylor Series: Centered at any point 'a'. Maclaurin Series: Centered specifically at x=0.

Taylor Series: Infinite sum of terms. Taylor Polynomials: Finite sum of terms (partial sum).

$\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n$

$\sum_{n=0}^\infty \frac{x^n}{n!}$

$\sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!}$

$\sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!}$

$\sum_{n=0}^\infty x^n$

$\sum_{n=0}^\infty (-x)^n$

$\sum_{n=0}^\infty (-x)^{2n}$

$\sum_{n=0}^\infty x^{2n}$

$\sum_{n=0}^\infty (-1)^n \frac{x^{n+1}}{n+1}$

$\sum_{n=0}^\infty {a \choose n}x^n$