Question

Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.) en + e−n e2n − 3

Answer 1

It looks like the sequence might be

$a_n = \dfrac{e^n + e^{-n}}{e^{2n - 3}}$

which we can simplify by dividing through by $e^n$ to get

$a_n = \dfrac{1 + e^{-2n}}{e^{n - 3}} = \dfrac1{e^{n-3}} + \dfrac1{e^{3n - 3}}$

As $n\to\infty$, both terms, and hence the given sequence will converge to 0.