Best possible bounds and monotonicity of segments of harmonic series. II. (English) Zbl 1007.40002

Simić, Slavko

Mat. Vesn. 50, No. 1-2, 5-10 (1998).

[For Part I see S. Simić, Mat. Vesn. 38, 331-336 (1986; Zbl 0648.40002).]
The author deals with sums \(s_n(p,q,A,B)=1/(np+A)+1/(np+A+1)+\cdots+1/(nq+B)\), where \(q>p>0\) and \(B\geq A-1\geq 0\) are integers. Setting \(m=p(2B+1)-q(2A-1)\), he proves that the sequence \(s_n(p,q,A,B)\) is strictly increasing for \(m\leq 0\). Also, in the case \(m>0\), the author finds \(n_0=n_0(m,p,q,A)\) such that for \(n\leq n_0\) the sequence \(s_n\) is strictly increasing and for \(n\geq n_0\) it is strictly decreasing. More general sums, where \(n\) is replaced by \(f(n)\), \(f\) strictly increasing, are also considered.

Reviewer: Miloš Arsenović (Beograd)

MSC:

40A05

Convergence and divergence of series and sequences

Citations:

Zbl 0648.40002