Following Tristan Needham, we are going to study two rational functions more to have a better understanding of the convergence of the power series, what he has called "The Mistery of Real Power Series".
We know that many real functions F(x) can be expressed (e.g. via Taylor's Theorem) as power series, and that this infinte series will normally only converge to F(x) in some interval of convergence -R < x < R. But how is R (the radius of convergence) determined by F(x)?.
Needham wrote: "It turns out that this question has a beautifully simple answer, but only if we investigate it in the complex plane. If we instead restrict ourselves to the real line -as mathematicians were forced to in the era in which such series were first employed- then the relationship between R and F(x) is utterly mysterious. Historically, it was precisely this mystery that led Cauchy to several of his breakthroughs in complex analysis." (Tristan Needham, Visual Complex Analysis, p. 64)
Consider the rational function
Its series, centered at the origin, has the interval of convergence (-1, 1).
It is easy to understand the interval of convergence of the series because the points x = 1 and x = -1 are singularities of the function itself. They are places where the module of the function becomes infinite.
This behavior is similar to other functions with singularities.
If we change the center of the power series, the radius of the interval of convergence is the distance from the center of the series to the nearest singularity. Singularities act like barriers.
We can compare this behavior with that of another function with a more mysterious behavior: Taylor polynomials (7): Rational function without real singularities.
Tristan Needham - Visual Complex Analysis. (pags. 64-77) - Oxford University Press