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#math/calculus
if Series#Geometric series, 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 : 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 of the graph of f at (x_0, f(x_0)): that is, the improved guess is the unique root of the 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.