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The ratio between the tail of a series and its approximating integral. (English) Zbl 1053.26011

Summary: For a strictly positive function \(f(x)\), let \(S(n)= \sum^\infty_{k=n} f(k)\) and \(I(x)=\int^\infty_x f(t)dt\) assumed convergent. If \(f'(x)/f(x)\) is increasing, then \(S(n)/I(n)\) is decreasing and \(S(n+1)/I(n)\) is increasing. If \(f''(x)/f(x)\) is increasing, then \(S(n)/I(n-\frac 12)\) is decreasing. Under suitable conditions, analogous results are obtained for the “continuous tail” defined by \(S(x)= \sum^\infty_{n=0} f(x+n)\): these results apply, in particular, to the Hurwitz zeta function.

MSC:

26D15 Inequalities for sums, series and integrals
26A48 Monotonic functions, generalizations
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