Question #158012

Find limit superior and limit inferior for the sequence (an)n∈N=((1/n)+(-1)^n)n∈N


1
Expert's answer
2021-01-29T05:10:57-0500

{an}={1n+(1)n}\{a_n\}=\{\frac{1}{n}+(-1)^n\}

an=1n+(1)n2|a_n|=|\frac{1}{n}+(-1)^n|\leq2 for all integers n1n\geq1. Hence {an}\{a_n\} is bounded.

The first few terms are {0,32,23,54,45,76,}\{0,\frac{3}{2},\frac{-2}{3},\frac{5}{4},\frac{-4}{5},\frac{7}{6},\cdots\}. The subsequence {a2n}={32,54,76,}\{a_{2n}\}=\{\frac{3}{2},\frac{5}{4},\frac{7}{6},\cdots\} converges to 1 and the subsequence {a2n1}={0,23,43,}\{a_{2n-1}\}=\{0,\frac{-2}{3},\frac{-4}{3},\cdots\} converges to -1. Hence S={1,1}S=\{-1,1\}

lim infan=inf(S)=1lim supan=sup(S)=1\liminf a_n=\inf(S)=-1\\ \limsup a_n=\sup(S)=1


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