Consider the sequence an=(1+1n)n2a_n = (1 + \frac{1}{n})^{n^2}an=(1+n1)n2. Does the series ∑n=1∞an\sum_{n=1}^{\infty} a_n∑n=1∞an converge?
Converges because limn→∞an=0\lim_{n \to \infty} a_n = 0limn→∞an=0
Diverges because limn→∞an≠0\lim_{n \to \infty} a_n \neq 0limn→∞an=0
Converges by the Root Test
Converges by the Comparison Test