Given the series ∑n=1∞an\sum_{n=1}^{\infty} a_n∑n=1∞an where an=∫nn+1e−x2dxa_n = \int_{n}^{n+1} e^{-x^2} dxan=∫nn+1e−x2dx. Is it convergent?
Converges by Comparison Test with ∫1∞e−x2dx\int_1^{\infty} e^{-x^2} dx∫1∞e−x2dx
Diverges because limn→∞an≠0\lim_{n \to \infty} a_n \neq 0limn→∞an=0
Diverges by the Integral Test
Converges because an<e−n2a_n < e^{-n^2}an<e−n2