If at a point $c$, a function $f$ has a power series expansion
                    $f\left(x\right)\=\sum _{n\=0}^{\infty }{a}_n{\left(x-c\right)}^n$
the coefficients $a_n$ are given by:
                   $a_n\=\frac{f^{\left(n\right)}\left(c\right)}{n\!}$

where $f^{\left(n\right)}\left(c\right)$ is the nth derivative of $f$ evaluated at the point $c$.  Named after the English mathematician Brook Taylor, this infinite series is called the Taylor expansion of the function $f$ at $c$.

The Taylor expansion of the exponential function ${\ⅇ}^x$ is:
                  ${\ⅇ}^x\=\sum _{n\=0}^{\infty }\frac{x^n}{n\!}$

from which it follows that:

                    $\ⅇ\=\sum _{n\=0}^{\infty }\frac{1}{n\!}$

Taylor approximations require both an expression and a point around which to expand.

Thus, around the point $x\=1$ the polynomial $x^7-5x^5\+4x^4-7x^2\+3$ behaves like $x^3-15x^2\+11x-1$.

The derivative of the arctan function has singularities at $I$ and $-I$. Therefore, the radius of convergence of the Taylor approximation around the origin is $1$.

You can compute and visualize Taylor approximations using the TaylorApproximationTutor command.

The FunctionChart routine plots a function and shows regions of positive and negative sign, increasing and decreasing, and positive and negative concavity.  By default:

1.  Roots are marked by circles.
2.  Extreme points are marked by diamonds.
3.  Inflection points are marked by crosses.
4.  Regions of increase and decrease are marked by red and blue lines, respectively.

5.  Regions of positive and negative concavity are marked by azure and purple fill, respectively, with arrows pointing in the direction of the concavity.

You can also perform curve analysis using the CurveAnalysisTutor command.

Given a point $a$ and an expression $f\left(x\right)$, the $x$-intercept of the tangent line through ($a$, $f\left(a\right)$) can be used as an approximation to a root of the expression $f\left(x\right)$.  The equation of the tangent line is:

Solving for zero:

As an example, consider the function $F\left(x\right)\=x^2-1$ and an initial point $x\=2.0$.

Repeating this another $9$ times:

The routine NewtonsMethod performs the same process.

The root to which a sequence of Newton iterations converges to depends on the initial point.  For example,

In general, when the root is not a double root, Newton's method is very efficient.  In the following example with Digits set to $30$, Newton's method converges to the root after only $7$ iterations.

You can also learn about Newton's method using the NewtonsMethodTutor command.