How do we know if a particular function can be represented as a power series? And once we have come up with a power series representation, how does one figure out its radius of convergence ?
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$\begingroup$A function can be represented as a power series if and only if it is complex differentiable in an open set. This follows from the general form of Taylor's theorem for complex functions.
Being real differentiable--even infinitely many times--is not enough, as the function $e^{-1/x^2}$ on the real line (equal to 0 at 0) is $C^\infty$ yet does not equal its power series expansion since all its derivatives at zero vanish. The reason is that the complexified version of the function is not even continuous at the origin.
$\endgroup$ 2 $\begingroup$To your question regarding radius of convergence, Wikipedia gives a good answer.
$\endgroup$ $\begingroup$This is a very general question, as one can create all sorts of power series for different functions. (e.g. Taylor series, Laurent series, Fourier series).
To give the obvious example of Taylor series: a power series representation of a function can be found if the function is infinitely differentiable in the neighbourhood of the given point.
With all power series, you will need to find the recursion relation (formula giving a successive term from the current term) and then use the ratio test to solve for the value of the input variable that gives a ratio of convergence of 1.
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