🔍Power series can extend functions beyond their original domain.
🔄Deleting terms from a power series can cause uncontrollable growth near the boundary.
⚠️Analytic continuation is a necessary condition for a well-behaved extension of a function.
🌀The Gap series, a modified power series, exhibits chaotic behavior near the boundary and has no analytic continuation.
🌐Analytic continuation allows functions to be extended in a more general and robust manner.