Étudier la convergence des séries \(\sum u_n\) suivantes:
1
\(u_n=\frac{n}{n^3+1}\)
2
\(u_n=\frac{\sqrt{n}}{n^2+\sqrt{n}}\)
3
\(u_n=n\sin\Big(\frac{1}{n}\Big)\)
4
\(u_n=\frac{1}{\sqrt{n}}\ln\Big(1+\frac{1}{\sqrt{n}}\Big)\)
5
\(u_n=\frac{\sqrt{n+1}-\sqrt{n}}{n}\)
6
\(u_n=\frac{(-1)^n+n}{n^2+1}\)
7
\(u_n=\frac{1}{n!}\)
8
\(u_n=\ln\left(\frac{n^2+n+1}{n^2+n-1}\right)\)