Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics II / 1 / c / i

At even integers $n=2k$, the terms are $1+1/(2k)\to 1$. At odd integers they are $-1-1/(2k-1)\to -1$. The largest eventual values therefore come from the subsequence at even integers, andExplicitly, every sufficiently late term is below $1+\epsilon$, while infinitely many even integers give terms greater than $1-\epsilon$.