Sorry for the delay. Grading.
Now we have power series expansions of functions around various points, and within various radii of convergence. We even have formulas to relate expansions about nearby points. But when we move from one point to a nearby point, the resulting series is only guaranteed to converge in a disk contained within the original disk. But then moving back to the original point we are only guaranteed convergence in an even smaller disk. Something seems amiss here.
Let’s look closely at the power series expansion about a given point :
converging for . We know that this function has a derivative, which again has a power series expansion about :
converging in the same radius. And so on, we find arbitrarily high derivatives
Now, we can specialize this by evaluating at the central point to find . That is, we have the formula for the power series coefficients:
This formula specifies the sequence of coefficients of a power series expansion of a function about a point uniquely in terms of the derivatives of the function at that point. That is, there is at most one power series expansion of any function about a given point.
So in our first example, of moving away from a point and back, the resulting series has the same coefficients we started with. Thus even though we were only assured that the series would converge in a much smaller disk, it actually converges in a larger disk than our formula guaranteed. In fact, this happens a lot: moving from one point to another we actually break new ground and “analytically continue” the function to a larger domain.
That is, we now have two overlapping disks, and each one contains points the other misses. Each disk has a power series expansion of a function. These expansions agree on the parts of the disks that overlap, so it doesn’t matter which rule we use to compute the function in that region. We thus have expanded the domain of our function by choosing different points about which to expand a power series.