#math/calculus 

if [[Series#Geometric series|geometric]], find $a$ and $r$, and use the formulas.

if algebraic:
- try getting the partial fraction of the function $a_n$
- begin to compute the series from beginning
- see whether any element of the partial fraction cancels any other iteration.

Newton's method ::: $x_n=x_n-1-f(x_1)/f'(x_1)$
Approximation the zero of a function.
[Newton's method - Wikipedia](https://en.wikipedia.org/wiki/Newton%27s_method) :
If the function satisfies sufficient assumptions and the initial guess is close, then
$$
x_{1}=x_{0}-\frac{f\left(x_{0}\right)}{f^{\prime}\left(x_{0}\right)}
$$ 
is a better approximation of the root than $x_0$. Geometrically, $(x_1, 0)$ is the intersection of the _x_-axis and the [tangent](https://en.wikipedia.org/wiki/Tangent "Tangent") of the [graph](https://en.wikipedia.org/wiki/Graph_of_a_function "Graph of a function") of $f$ at $(x_0, f(x_0))$: that is, the improved guess is the unique root of the [linear approximation](https://en.wikipedia.org/wiki/Linear_approximation "Linear approximation") at the initial point. The process is repeated as$$
x_{n+1}=x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)}
$$ functions and to systems of equations.