This might seem super easy, but could anyone give me some hint how to show the following An is convergent?

1.n!/n^n

2.(1-1/n)^(n^2)

3. 1-cos (1/n)

I've tried the limit test/ratio test, but not working. Maybe comparison test?

Also, could anyone help calculating the series using taylor expansion of cos?

the sum of all (cos n)/(2^n) where n goes from 0 to infinity?

Any help is greatly appreciated!