Geodesic
On the Convergence Rate of a Recursively Defined Sequence
Matematičeskie zametki, Tome 93 (2013) no. 2, pp. 195-201

Voir la notice de l'article provenant de la source Math-Net.Ru

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/}
}
TY  - JOUR
AU  - Jong-Yi Chen
AU  - Yunshyong Chow
TI  - On the Convergence Rate of a Recursively Defined Sequence
JO  - Matematičeskie zametki
PY  - 2013
SP  - 195
EP  - 201
VL  - 93
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2013_93_2_a3/
LA  - ru
ID  - MZM_2013_93_2_a3
ER  - 
%0 Journal Article
%A Jong-Yi Chen
%A Yunshyong Chow
%T On the Convergence Rate of a Recursively Defined Sequence
%J Matematičeskie zametki
%D 2013
%P 195-201
%V 93
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2013_93_2_a3/
%G ru
%F 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/