The Ratio Between the Tail of a Series and its Approximating Integral

Volume 4, Issue 2, 2003

Article 25

THE RATIO BETWEEN THE TAIL OF A SERIES AND ITS APPROXIMATING INTEGRAL

DEPARTMENT OF MATHEMATICS AND STATISTICS,
LANCASTER UNIVERSITY
LANCASTER LA1 4YF,
GREAT BRITAIN.
E-Mail: g.jameson@lancaster.ac.ukURL: http://www.maths.lancs.ac.uk/~jameson/

Received 20 September, 2002; Accepted 10 February, 2003.
Communicated by: A. Fiorenza

ABSTRACT.

For a strictly positive function , 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 is decreasing and 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.

Key words:
Series, Tail, Ratio, Monotonic, Zeta function.

2000 Mathematics Subject Classification:
26D15, 26D10, 26A48.

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