Prove whether the following series is convergent or divergent Prove the convergence of the following series:
$$\sum\frac{1 + 2^n + 5^n}{3^n}$$
My idea was to write it as:
$$\frac{1}{3^n} + \frac{2^n}{3^n} + \frac{5^n}{3^n}$$
Which would mean that $\frac{1}{3^n}$ is a geometric series with a ratio of $\frac{1}{3}$, and since $\frac{1}{3} <1$,$\frac{1}{3^n}$ is convergent. However, I don't know what to do with the rest...

Simply write $\frac{5^n}{3^n}$ as $\left(\frac53\right)^n$. Then,
$$\sum \frac{1+2^n+5^n}{3^n}>\sum \frac{5^n}{3^n}=\sum\left(\frac53\right)^n>\sum 1^n=\infty$$
So the entire sum diverges.
