Answer
The series $\sum_{n=1}^{\infty} \frac{2^{n}+4^{n}}{7^{n}}$ converges.
Work Step by Step
In the series $\sum_{n=1}^{\infty} \frac{2^{n}+4^{n}}{7^{n}}$, we have the series
$$\sum_{n=1}^{\infty} \frac{2^{n} }{7^{n}}=\sum_{n=1}^{\infty} (\frac{2 }{7 })^n$$
which is a convergent geometric series with $r=2/7\lt 1$. Also, the series
$$\sum_{n=1}^{\infty} \frac{4^{n} }{7^{n}}=\sum_{n=1}^{\infty} (\frac{4 }{7 })^n$$
is a convergent geometric series with $r=4/7\lt 1$. Hence, the series $\sum_{n=1}^{\infty} \frac{2^{n}+4^{n}}{7^{n}}$ converges.
