Question 33
a) Show that the sequence $a_{n} = \int_{0}^{n} e^{-x^{2}} dx$, $n \in \mathbb{N}$, is convergent.
b) If $f:[0, \infty) \rightarrow [0, \infty)$ is a continuous function with $\lim_{x \rightarrow \infty} f(x) = 0$, does it necessarily follow that the sequence $x_{n} = \int_{0}^{n} f(x) dx$, $n \in \mathbb{N}$, is convergent? Justify.
b) If $f:[0, \infty) \rightarrow [0, \infty)$ is a continuous function with $\lim_{x \rightarrow \infty} f(x) = 0$, does it necessarily follow that the sequence $x_{n} = \int_{0}^{n} f(x) dx$, $n \in \mathbb{N}$, is convergent? Justify.