Geodesic
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. 
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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/

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