AP Calculus BC

Does the Sum of n!/n^n Converge? Yes

The sum of n factorial over n to the n converges. The ratio test gives a limit of 1 over e, about 0.368, which is less than 1. This is the classic case where the ratio limit turns out to be a genuinely interesting constant rather than 0 or infinity.

∑n=1∞n!nn\sum_{n=1}^{\infty}\frac{n!}{n^{n}}

Converges

Settled by the ratio test.

The ratio produces the definition of e

an+1an=(n+1)!(n+1)n+1⋅nnn!=nn(n+1)n=1(1+1n)n\frac{a_{n+1}}{a_{n}} = \frac{(n+1)!}{(n+1)^{n+1}}\cdot\frac{n^{n}}{n!} = \frac{n^{n}}{(n+1)^{n}} = \frac{1}{\left(1+\frac{1}{n}\right)^{n}}

That denominator is the classic limit defining ee, so the ratio limit is its reciprocal.

lim⁡n→∞(1+1n)n=e  ⟹  L=1e≈0.368<1\lim_{n \to \infty}\left(1+\frac{1}{n}\right)^{n} = e \implies L = \frac{1}{e} \approx 0.368 < 1

n^n outgrows n!

A ratio limit strictly below 1 means the denominator wins, so n^n grows faster than n factorial. That is the last link in the growth ordering, and this series is where it gets proved.

The mistakes students make

  • Simplifying (n+1)n+1(n+1)^{n+1} to (n+1)n(n+1)^{n} without accounting for the extra factor. It is (n+1)n⋅(n+1)(n+1)^{n} \cdot (n+1), and that spare (n+1)(n+1) cancels the (n+1)(n+1) from the factorial.
  • Missing the definition of ee and leaving the answer as an unevaluated limit.
  • Assuming n!n! beats nnn^{n} because a factorial is a huge number. Here it loses.

Not sure which test a series wants?

The Convergence Test Chooser walks the decision in order: nth term first, then geometric and p-series pattern matching, then alternating structure, then the ratio test, and finally the comparison family.

Frequently asked questions

Does the sum of n!/n^n converge?

Yes. The ratio test gives L=1e≈0.368<1L = \frac{1}{e} \approx 0.368 < 1.

Where does e come from?

The ratio simplifies to 1(1+1/n)n\frac{1}{(1+1/n)^{n}}, and that inner limit is the definition of ee.

Which grows faster, n! or n^n?

nnn^{n}. A ratio limit below 11 is exactly the statement that the denominator wins.