Module series_api 

Summation, products, series and formal-power-series methods on Ex. Summation, products, and series extension methods on Ex.

• Ex::summation / Ex::try_summation — symbolic Σ
• Ex::product_over / Ex::try_product_over — symbolic Π
• Ex::hypergeometric_ratio — the term ratio t(k+1)/t(k)
• Ex::is_absolutely_convergent — absolute convergence of Σ t(k)
• Ex::series_at_infinity / Ex::series_at_neg_infinity — asymptotic expansions
• Ex::series_dir / Ex::try_series_dir — one-sided expansions

What a series is

Ex::series and the methods here compute the asymptotic expansion as var → point through real values (two-sided for series, one-sided for Ex::series_dir and at ±∞), in the same sense as Ex::limit: every term of exponent < order is present and exact, and the remainder is O((var − point)^order) along the reals. It is not a Laurent series over the complex plane: exp(−1/x²) expands to 0 at 0 (SymPy: O(x**4)), exp(−1/x) has no two-sided expansion there (0 from the right, none from the left). One-sided expansions from the right and those at ±∞ may contain logarithms (ln Γ(x) at ∞ is Stirling’s series, Ei(x) at 0⁺ is γ + ln x + x + …). Where no such expansion exists or it cannot be established, a formal Series node is returned (try_* variants: Err). Use Ex::residue for residues: a series is not a Laurent expansion at an essential singularity.