We’ve rearranged the terms from a series for log 2 and obtained half the original value. Until the early 19th century, the evaluation of infinite series was not troubled by rearrangement issues as above, since there wasn’t a clear distinction between two issues: defining convergence of a series and computing a series. An infinite series needs a precise defining algorithm, such as taking a limit...

The principle of analytic continuation makes it possible to calculate effectively the corresponding convergent power series about all points where a holomorphic continuation exists. However, the nature of the function near its singularities cannot be so readily deduced from the series (1.1). Often series expansions about such singular points do exist, and sometimes it is possible to calculate t...

We show that the iterative logarithm of the power series ez − 1 is differentially transcendental over the ring of convergent power series.

Given function f with an absolutely convergent Fourier series,