Mathematics

Mathematics

Problem

Let ff be a continuous, positive, and decreasing function on [1,∞)[1,\infty). Which of the following inequalities is always true? a) f(1)≤∑n=1∞f(n)≤∫1∞f(x) dx\displaystyle f(1)\le \sum_{n=1}^{\infty} f(n) \le \int_{1}^{\infty} f(x)\,dx b) ∫1∞f(x) dx≤∑n=1∞f(n)≤f(1)+∫1∞f(x) dx\displaystyle \int_{1}^{\infty} f(x)\,dx \le \sum_{n=1}^{\infty} f(n) \le f(1)+\int_{1}^{\infty} f(x)\,dx c) ∫2∞f(x) dx≤∑n=1∞f(n)≤∫1∞f(x) dx\displaystyle \int_{2}^{\infty} f(x)\,dx \le \sum_{n=1}^{\infty} f(n) \le \int_{1}^{\infty} f(x)\,dx d) f(1)+∫1∞f(x) dx≤∑n=1∞f(n)≤∫1∞f(x) dx\displaystyle f(1)+\int_{1}^{\infty} f(x)\,dx \le \sum_{n=1}^{\infty} f(n) \le \int_{1}^{\infty} f(x)\,dx

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