Testing Σ(n!/nⁿ)

Calculus 2 · Axiom Academy

Determine whether the series converges using the ratio test and properties of limits Determine whether the series converges or diverges. Because the terms mix a factorial with an exponential, the ratio test is the natural tool. Excellent work! You've applied the ratio test to a series mixing a factorial and an exponential. Here's what we learned: Ratio Test Setup: For a series , compute . If , the series converges. Factorial Simplification: Remember that , which lets the factorials cancel in the ratio. Power Manipulation: Split a power before cancelling: . The e Connection: The limit is fundamental. Here we used . Convergence Conclusion: Since , the series converges absolutely. The ratio test is especially powerful for series involving factorials and exponentials. Master the algebraic manipulation, and even intimidating convergence problems become routine.

This is the written version of the interactive lesson above. See the full Calculus 2 course.