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sum of digits in base ten, limit of sequence

Source: VJIMC 1996 2.3

October 19, 2021
real analysislimitsnumber theory

Problem Statement

Let cif⁡(x)\operatorname{cif}(x) denote the sum of the digits of the number xx in the decimal system. Put a1=199719961997a_1=1997^{1996^{1997}}, and an+1=cif⁡(an)a_{n+1}=\operatorname{cif}(a_n) for every n>0n>0. Find lim⁡n→∞an\lim_{n\to\infty}a_n.
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