Q:

Show that the sequence $a_n = \frac{2^n+\cos{(n\pi)}3^n}{5^n}$ converges to $0$.

Show that the sequence $a_n = \frac{2^n+\cos{(n\pi)}3^n}{5^n}$ converges to $0$.
Would we use the squeeze theorem in this case? Please help.

A:

This is how would I prove it:
$$-1 \leq \cos (n \pi) \leq 1$$
When
$\cos(n \pi)=-1$,
$$\lim_{n \to \infty}\frac{2^n-3^n}{5^n}=0$$
as $2^n-3^n \leq 5^n$, when $n \geq -1$ and both $2^n-3^n$ amd $5^n$ are motonically increasing.
When
$\cos(n \pi)=1$,
$$\lim_{n \to \infty}\frac{2^n+3^n}{5^n}=0$$
as $2^n+3^n \leq 5^n$, when $n \geq 1$ and both $2^n+3^n$ amd $5^n$ are motonically increasing.
So,
$$\lim_{n \to \infty}\frac{2^n+ \cos(n \pi)3^n}{5^n}=0$$
This is in no way rigorous, just my thoughts.  

