Proving convergence of factorial w/o Ratio Test

[mvpshaq32]

Homework Statement

Determine whether 1/n! diverges or converges.
So far, we have only learned the comparison tests, p-series, geometric series, divergence test, and integral test, so I can only use these tests to prove it.

Homework Equations

N/a

The Attempt at a Solution

I thought about using limit comparison with my b_n=1/n^n, but I can't determine if that converges or not, so I don't know what to do.

 

[vela]

Why did you choose $b_n = 1/n^n$? Why not a simpler series that you know converges or diverges?

 

[flyingpig]

The Comparison Test is your friend

 

[HallsofIvy]

It should be clear that for n> 3, n^2> n!.

 

[Maxter]

It should be clear that for n> 3, n^2> n!.

I think you meant 2^n < n! ? Easy typo to make.

Oops, it's 2^(n-1) < n!

 

[TPAINE]

It's monotonic, so if you show that it is bounded, you're in business. Now by the comparison test, we have n!>n^2 for sufficiently large n, so take reciprocals and go from there.