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complicated limit of sequence, a_(n+2)=a_(n+1)+a_n/2^n

Source: VJIMC 2003 2.3

July 13, 2021

limitsreal analysisSequences

Problem Statement

Let

\{a_n\}^\infty_{n=0}

be the sequence of real numbers satisfying

a_0=0

a_1=1

and

a_{n+2}=a_{n+1}+\frac{a_n}{2^n}

for every

n\ge0

. Prove that

\lim_{n\to\infty}a_n=1+\sum_{n=1}^\infty\frac1{2^{\frac{n(n-1)}2}\displaystyle\prod_{k=1}^n(2^k-1)}.