In Response To 'Re: bug in Series function'
Let's try this again, without the numeric computations. See attached notebook.
Note that by definition, S[z] is not a rational function
(* the exponents defined by g[n] will skip an integer when it runs across a prime; when this happens the power series has zero as coefficient; these zeros occur irregularly and don't allow the coefficients to satisfy a linear recurrence *)
in particular, S[z] does not equal T[z] although the
latter is a majorant for the former.
Yet the Series function claims that S[z]=T[z] as power series!
Note: if we change the defining sum for S[z] to run from 1 to 1000, say, then Series gets it right. But going from 1 to Infinity results in an error.
Attachment: series_bug.nb, URL: ,