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

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

$\frac{f^{(n)}(a)}{n!}(x-a)^n$

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

1+5x+\frac{25}{2}x^2+\frac{125}{6}x^3

An infinite sum of terms expressed in terms of the function's derivatives at a single point.

A Taylor series centered at $x=0$.

The $n^{\text{th}}$ derivative of the function $f(x)$ evaluated at $x=a$.

The $n^{\text{th}}$ partial sum of the Taylor series.

Using a Taylor polynomial to estimate the value of a function at a specific point.

To approximate the value of a function using a polynomial, especially useful when the function is difficult to compute directly.

Because they are centered at $x=0$, which often simplifies calculations.

The function $f(x)$ evaluated at $x=a$, i.e., $f(a)$.

Generally, a higher degree polynomial provides a more accurate approximation, especially closer to the center $x=a$.

A Taylor polynomial is a truncated Taylor series, taking only the first few terms of the infinite series.

Derivatives determine the coefficients of the polynomial terms, reflecting the function's rate of change at the center.