What does interval of convergence for power series show? For the simplest case, $f(x)=\dfrac1{1-x}$. This can be represented by $\sum\limits_{i=1}^nx^{n-1}$ or $\sum\limits_{i=0}^nx^n$. The series is only equal to that value for $|x|< 1$, so the interval of convergence is $-1<x<1$.
What exactly does that mean? Does it mean that the function is only approximated by the series for those specific values of $x$?
E.g.: For $x=5$, the series does not apply? The series will equal $1 + 5 + 125 + 625 + \cdots$ while $f(5) = -\dfrac14$. Is my understanding correct?
Also, what is the motivation behind this operation? Why convert a function into a series in the first place?
 A: An entire series is a very powerful way to represent functions. Like polynomials, it only involves additions and multiplications. The term-wise differentiation and integrations are straightforward, as is numerical evaluation (provided the convergence speed is good enough).
They have many properties in common, so there is a complete theory of entire functions. Even though they only converge inside their radius, they can be extended elsewhere by changing the center and recomputing the coefficients. The radius of convergence usually corresponds to singular points in the function (function going to infinity), that cannot be "crossed" by the series representation.
They are also a general way to study the behavior of smooth functions around the origin.
A: Yes that is correct. Especially your example of $f(5) = -1/4$ shows that the function wouldn't equal its series expansion in this case.
Why use series representation? Easy to integrate and differentiate a series. May be computationally efficient to approximate a function by a series. You could use the series to understand limits and other properties of a function.
