Does nsin(2πen!) have a predictable pattern for convergence?

[cragar]

Homework Statement

Does $nsin(2\pi en!)$
converge and if so what does it converge to.
this is a sequence and n is a positive integer.

The Attempt at a Solution

My teacher gave us a hint and to write e in a Taylor series.
$nsin(2\pi (1+1+\frac{1}{2!}+\frac{1}{3!}...)n!)$
so we could multiply the n! through to the e stuff. And then look for a pattern their to see what it converges to.

 

[Dick]

Homework Statement

Does $nsin(2\pi en!)$
converge and if so what does it converge to.
this is a sequence and n is a positive integer.

The Attempt at a Solution

My teacher gave us a hint and to write e in a Taylor series.
$nsin(2\pi (1+1+\frac{1}{2!}+\frac{1}{3!}...)n!)$
so we could multiply the n! through to the e stuff. And then look for a pattern their to see what it converges to.

Well, what are your thoughts? I see a bunch of stuff that is a multiple of 2pi in the sin function. But I also see a bunch of stuff that's not.