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 .

- If
\displaystyle{ \sum_{n=1}^{\infty} a_n } diverges, then\displaystyle{\int_{1}^{\infty} a(x) \, dx } choose converges diverges cannot be determined **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 .

- If
\displaystyle{ \sum_{n=1}^{\infty} a_n } converges, then\displaystyle{\int_{1}^{\infty} a(x) \, dx } choose converges diverges cannot be determined **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 .

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