Mathematics

Mathematics

Problem

Which set of conditions must a function ff satisfy so that the convergence of ∑n=1∞f(n)\displaystyle\sum_{n=1}^{\infty} f(n) can be determined by comparing it to ∫1∞f(x) dx\displaystyle\int_{1}^{\infty} f(x)\,dx? a) ff is positive, continuous on [1,∞)[1,\infty), and f′(x)<0f'(x)<0 for all x>1x>1. b) ff is positive, continuous on [1,∞)[1,\infty), and decreasing on (1,∞)(1,\infty). c) ff is non-negative, continuous on [1,∞)[1,\infty), and lim⁡x→∞f(x)=0\displaystyle\lim_{x\to\infty}f(x)=0.},{ d) ff is positive and continuous on [1,∞)[1,\infty).

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