Power series can be identified as closed-form functions by algebraically manipulating them into the form of known series (such as the exponential series or the geometric series), using techniques analogous to polynomial manipulation: factoring out common terms, splitting a sum into constituent sub-series, adding and subtracting terms to align a series' starting index with a known series' standard form, and re-indexing (shifting the summation variable) to match a standard series' exponent structure. This belongs to the theory of power series within single-variable calculus, connecting formal series manipulation to the closed-form functions (e.g., e^x, 1/(1-x)) that those series represent within their interval of convergence.
Power series can be identified as closed-form functions by algebraically manipulating them into the form of known series (such as the exponential series or the geometric series), using techniques ana…