Suppose $a(x)$ is a continuous, positive, decreasing function for $x$ in the interval $\lbrack 1, \infty)$, and $\lbrace a_n \rbrace$ is the sequence defined by $a_n = a(n)$ for every natural number $n$.

1. If $\displaystyle{ \sum_{n=1}^{\infty} a_n }$ diverges, then $\displaystyle{\int_{1}^{\infty} a(x) \, dx }$ choose converges diverges cannot be determined because
   A. $\displaystyle \int_{0}^{\infty} a(x) \, dx < \sum_{n=1}^{\infty} a_n$.
   B. $\displaystyle \int_{1}^{\infty} a(x) \, dx < \sum_{n=1}^{\infty} a_n$.
   C. $\displaystyle \sum_{n=1}^{\infty} a_n < \int_{1}^{\infty} a(x) \, dx$.
   D. $\displaystyle \sum_{n=1}^{\infty} a_n < a_1 + \int_{1}^{\infty} a(x) \, dx$.
   
2. If $\displaystyle{ \sum_{n=1}^{\infty} a_n }$ converges, then $\displaystyle{\int_{1}^{\infty} a(x) \, dx }$ choose converges diverges cannot be determined because
   A. $\displaystyle \int_{1}^{\infty} a(x) \, dx < \sum_{n=1}^{\infty} a_n$.
   B. $\displaystyle \int_{0}^{\infty} a(x) \, dx < \sum_{n=1}^{\infty} a_n$.
   C. $\displaystyle \sum_{n=1}^{\infty} a_n < a_1 + \int_{1}^{\infty} a(x) \, dx$.
   D. $\displaystyle \sum_{n=1}^{\infty} a_n < \int_{1}^{\infty} a(x) \, dx$.