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Sequence

Source: 22nd Hellenic Senior mo , 2005 , prb2.

November 17, 2005
integrationalgebra proposedalgebra

Problem Statement

The sequence (an)(a_n) is defined by a1=1a_1=1 and an=an−1+1n3a_n=a_{n-1}+\frac{1}{n^3} for n>1.n>1. (a) Prove that an<54a_n<\frac{5}{4} for all n.n. (b) Given ϵ>0\epsilon>0, find the smallest natural number n0n_0 such that ∣an+1−an∣<ϵ{\mid a_{n+1}-a_n}\mid<\epsilon for all n>n0.n>n_0.
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