Is ∑n=1∞nn+1\sum_{n=1}^{\infty} \frac{n}{n+1}∑n=1∞n+1n convergent?
Yes
No, limn→∞an=1≠0\lim_{n \to \infty} a_n = 1 \neq 0limn→∞an=1=0.
Yes, by Comparison Test.
No, by Ratio Test.