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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:

26D15,26D10,26A48.

  
  Editors: Alberto Fiorenza,  
 
  Abstract:

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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