Some relations connecting measures of convergence for series with non- increasing terms. (English) Zbl 0601.40002
Some results due to J. B. Wilker [Can. Math. Bull. 21, 237-240 (1978; Zbl 0401.57024)] associated with the rate of convergence of series \(\sum^{\infty}_{n=1}a_ n\) are improved. The existence of convergent series \(\sum^{\infty}_{n=1}a_ n\) for which \(a_ n\downarrow 0\), \(\liminf \log a_ n/\log (n^{-1})=1,\quad \limsup \log a_ n/\log (n^{-1})=\infty\) and
\[
\lim (\log (\sum_{j>n}a_ j)/\log (n^{- 1}))=\lim (\log (\sum_{j>n}a_ j)/\log a_{n+1})=0
\]
is proved.
MSC:
| 40A05 | Convergence and divergence of series and sequences |
Keywords:
Olivier’s theoremCitations:
Zbl 0401.57024References:
| [1] | DOI: 10.1112/plms/s2-11.1.411 · JFM 43.0312.01 · doi:10.1112/plms/s2-11.1.411 |
| [2] | DOI: 10.2307/2371946 · Zbl 0035.03901 · doi:10.2307/2371946 |
| [3] | DOI: 10.1007/BF02018498 · Zbl 0371.52006 · doi:10.1007/BF02018498 |
| [4] | Knopp, Theory and application of Infinite Series (1951) · Zbl 0042.29203 |
| [5] | Bromwich, Introduction to the theory of infinite series (1926) |
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