1. Find derivatives of $f(x)$. 2. Evaluate derivatives at $x=a$. 3. Plug values into Taylor series formula. 4. Simplify.

1. Find derivatives of $f(x)$. 2. Evaluate derivatives at $x=0$. 3. Plug values into Maclaurin series formula. 4. Simplify.

1. Find $f'(x)$, $f''(x)$, $f'''(x)$. 2. Evaluate at $x=1$. 3. Use the Taylor polynomial formula up to the third degree. 4. Simplify: $(x-1)-\frac{1}{2}(x-1)^2+\frac{1}{3}(x-1)^3$.

1. Find derivatives up to order 5. 2. Evaluate at $x=0$. 3. Use Maclaurin polynomial formula. 4. Simplify: $1 - \frac{x^2}{2} + \frac{x^4}{24}$.

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

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

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

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

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

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.