Integral Test Comparison

Question

Let $$f$$ be a positive, continuous, and decreasing function for $$x \ge 1$$. If $$\displaystyle \int_1^\infty f(x)\,dx = 5$$, which of the following statements about the series $$\displaystyle \sum_{n=1}^\infty f(n)$$ must be true?

Accepted Answer

$$\displaystyle \sum_{n=1}^\infty f(n)$$ converges, and $$\displaystyle \sum_{n=1}^\infty f(n) > 5$$

Answer

$$\displaystyle \sum_{n=1}^\infty f(n) = 5$$

Answer

$$\displaystyle \sum_{n=1}^\infty f(n)$$ converges, and $$\displaystyle \sum_{n=1}^\infty f(n) < 5$$

Answer

$$\displaystyle \sum_{n=1}^\infty f(n) = 0$$

Integral Test Comparison

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