Mathematics

Mathematics

Problem

Suppose {ak}\{a_k\} is an infinite series with ak=f(k)a_k=f(k), where ff is continuous, positive, and decreasing on [1,∞)[1,\infty). Let Sn=∑k=1nakS_n=\sum_{k=1}^{n}a_k and Rn=∑k=n+1∞akR_n=\sum_{k=n+1}^{\infty}a_k. Which of the following inequalities always holds? a) ∫n∞f(x) dx<Rn<∫n+1∞f(x) dx\displaystyle\int_{n}^{\infty} f(x)\,dx < R_n < \int_{n+1}^{\infty} f(x)\,dx b) Rn=∫nn+1f(x) dx\displaystyle R_n = \int_{n}^{n+1} f(x)\,dx c) ∫n+1∞f(x) dx<Rn<∫n∞f(x) dx\displaystyle\int_{n+1}^{\infty} f(x)\,dx < R_n < \int_{n}^{\infty} f(x)\,dx d) ∫0nf(x) dx<Rn<∫n∞f(x) dx\displaystyle\int_{0}^{n} f(x)\,dx < R_n < \int_{n}^{\infty} f(x)\,dx

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