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  Volume 4, Issue 2, Article 25
The Ratio Between the Tail of a Series and its Approximating Integral

    Authors: Graham Jameson,  
    Keywords: Series, Tail, Ratio, Monotonic, Zeta function.  
    Date Received: 20/09/02  
    Date Accepted: 10/02/03  
    Subject Codes:


    Editors: Alberto Fiorenza,  

For a strictly positive function $ f(x)$, let $ S(n)=sum_{k=n}^{infty }f(k)$ and $ I(x)=int_{x}^{infty }f(t)dt$, assumed convergent. If $ f^{prime }(x)/f(x)$ is increasing, then $ S(n)/I(n)$ is decreasing and $ S(n+1)/I(n)$ is increasing. If $ f^{prime prime }(x)/f(x)$ is increasing, then $ S(n)/I(n-%% frac{1}{2})$ is decreasing. Under suitable conditions, analogous results are obtained for the ``continuous tail'' defined by $ S(x)=sum_{n=0}^{infty }f(x+n)$: these results apply, in particular, to the Hurwitz zeta function.

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