On the Convergence Rate of a Recursively Defined Sequence
Matematičeskie zametki, Tome 93 (2013) no. 2, pp. 195-201
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Consider the following recursively defined sequence:
$$
\tau_1 =1,\qquad \sum^n_{j=1} \frac{1}{\sum^n_{s=j}\tau_s}=1\quad \text{for}\quad n\geq 2,
$$
which originates from a heat conduction problem first studied by Myshkis (1997). Chang, Chow, and Wang (2003) proved that
$$
\tau_n = \log n +O(1) \qquad \text{for large}\quad n.
$$
In this note, we refine this result to
$$
\tau_n= \log n + \gamma+O \biggl(\frac{1}{\log n}\biggr).
$$
where $\gamma$ is the Euler constant.
Keywords:
difference equation, heat equation, asymptotic behavior, feedback control.
@article{MZM_2013_93_2_a3,
author = {Jong-Yi Chen and Yunshyong Chow},
title = {On the {Convergence} {Rate} of a {Recursively} {Defined} {Sequence}},
journal = {Matemati\v{c}eskie zametki},
pages = {195--201},
publisher = {mathdoc},
volume = {93},
number = {2},
year = {2013},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/MZM_2013_93_2_a3/}
}
Jong-Yi Chen; Yunshyong Chow. On the Convergence Rate of a Recursively Defined Sequence. Matematičeskie zametki, Tome 93 (2013) no. 2, pp. 195-201. http://geodesic.mathdoc.fr/item/MZM_2013_93_2_a3/