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Prove that the improper integral ∫ f(x) dx and ∑ f(n) both converge or both diverge

  1. Assume that f is a monotonically decreasing function for all x \geq 1 and that

        \[ \lim_{x \to +\infty} f(x) = 0. \]

    Prove that the improper integral and the series

        \[ \int_1^{\infty} f(x)  \, dx \qquad \text{and} \qquad \sum_{n=1}^{\infty} f(n) \]

    both converge or both diverge.

  2. Give a counterexample to the theorem in part (a) in the case that f is not monotonic, i.e., find a non-monotonic function f such that \sum f(n) converges but \int_1^{\infty} f(x) \, dx diverges.

    Incomplete.

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