Answer
The sequence $y_n$ converges to $0$.
Work Step by Step
We have
$$\lim_{n\to \infty}y_n=\lim_{n\to \infty}\frac{e^{n}+(-3)^{n}}{5^{n}}\\
=\lim_{n\to \infty}\frac{e^{n} }{5^{n}}+\frac{ (-3)^{n}}{5^{n}}\\
=\lim_{n\to \infty}(\frac{e }{5} )^{n}+(\frac{-3}{5})^{n}=0.$$
Here we used the facts that $|e/5|\lt 1$ and $|-3/5|\lt 1$.
Hence, by Theorem 1, the sequence $y_n$ converges to $0$.
