Determine whether the sequence \( \left\{a_{n}\right\} \) converges or diverges. If it converges, find its limit. (1) \( a_{n}=\frac{n !}{n^{n}} \) (2) \( a_{n}=\frac{(\ln n)^{\pi}}{\sqrt{n}} \) ((3) a
n
​
=
ln(n
2
+1)+1
ln(n+1)
​
(4) a
n
​
=n
2
(1−cos
n
1
​
)