Testing ∑n!/nⁿ

Real Analysis · Axiom Academy

EXAMPLE Testing with the Ratio Test Apply the ratio test to decide whether the series converges or diverges. Determine whether the series converges or diverges. The factorial in the numerator makes the ratio test the natural tool. What the ratio does as n grows Each dot is the term ratio aₙ₊₁/aₙ. The dots settle onto the dashed line at 1/e ≈ 0.368, which lies below the convergence threshold L = 1 — so the series converges. Nice work — you applied the ratio test end to end to prove convergence. Remember: The Ratio Test: if , the series converges. Factorial Simplification: is what lets the factorials cancel. Power Manipulation: recognizing exposes the limit. The limit 1/e : since , we get . Interpretation: , so the series converges absolutely. The ratio test is especially powerful for series built from factorials and exponentials — a technique you'll use throughout Real Analysis.

This is the written version of the interactive lesson above. See the full Real Analysis course.