MathDB

Problem 3

Part of 2003 VJIMC

Problems(2)

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

Source: VJIMC 2003 2.3

7/13/2021
Let {an}n=0∞\{a_n\}^\infty_{n=0} be the sequence of real numbers satisfying a0=0a_0=0, a1=1a_1=1 and an+2=an+1+an2na_{n+2}=a_{n+1}+\frac{a_n}{2^n}for every n≥0n\ge0. Prove that lim⁡n→∞an=1+∑n=1∞12n(n−1)2∏k=1n(2k−1).\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)}.
limitsreal analysisSequences
formalization of Ramanujan's result

Source: VJIMC 2003 1.3

7/12/2021
Find the limit lim⁡n→∞1+21+3…+(n−1)1+n.\lim_{n\to\infty}\sqrt{1+2\sqrt{1+3\sqrt{\ldots+(n-1)\sqrt{1+n}}}}.
limitsreal analysisNested Radicals