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

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 $1-\epsilon$.

The sine has values $1,0,-1,0$ on the four residue classes of $n$ modulo four. For $n=4k+1$, the exponential function gives $b_n=e^{1-1/n}\to e$; the other residue classes tend to $1,e^{-1},1$. All terms satisfy $b_n\le e$, and the first class approaches $e$. Hence