To evaluate the sum

$\sum_{n=1}^{\infty} \frac{3^n + 4^n}{5^n},$

we can split it into two separate sums:

$\sum_{n=1}^{\infty} \frac{3^n}{5^n} + \sum_{n=1}^{\infty} \frac{4^n}{5^n}.$

Each of these is a geometric series, which we can handle separately.

1. First Series: $\sum_{n=1}^{\infty} \frac{3^n}{5^n}$ $= \sum_{n=1}^{\infty} \left(\frac{3}{5}\right)^n.$ This is a geometric series with first term $a = \frac{3}{5}$ and common ratio $r = \frac{3}{5}$. For a geometric series $\sum_{n=1}^{\infty} ar^n$, the sum is given by: $\frac{a}{1 - r} = \frac{\frac{3}{5}}{1 - \frac{3}{5}} = \frac{\frac{3}{5}}{\frac{2}{5}} = \frac{3}{2}.$

2. Second Series: $\sum_{n=1}^{\infty} \frac{4^n}{5^n}$ $= \sum_{n=1}^{\infty} \left(\frac{4}{5}\right)^n.$ This is also a geometric series with first term $a = \frac{4}{5}$ and common ratio $r = \frac{4}{5}$. The sum of this series is: $\frac{a}{1 - r} = \frac{\frac{4}{5}}{1 - \frac{4}{5}} = \frac{\frac{4}{5}}{\frac{1}{5}} = 4.$

Adding these two results together:

$\sum_{n=1}^{\infty} \frac{3^n + 4^n}{5^n} = \frac{3}{2} + 4 = \frac{3}{2} + \frac{8}{2} = \frac{11}{2}.$

Thus, the sum is:

$\frac{11}{2}.$

Would you like further details on any step?

Here are some related questions for further practice:

1. How would you evaluate $\sum_{n=1}^{\infty} \frac{2^n + 3^n}{6^n}$?
2. What is the sum of an infinite geometric series where the first term is $\frac{5}{7}$ and the common ratio is $\frac{5}{7}$?
3. Can you solve $\sum_{n=1}^{\infty} \frac{n^2}{2^n}$ using different techniques?
4. How does the convergence of a geometric series change if the common ratio is negative?
5. How would you solve $\sum_{n=1}^{\infty} \frac{n}{3^n}$?

Tip: For a geometric series $\sum_{n=0}^{\infty} ar^n$, the series converges to $\frac{a}{1 - r}$ if $|r| < 1$.