How to find the sum of subsequences and sum of the series for series like these? $$\sum_{n=0}^\infty \frac{3^n+4^n}{(-7)^n}$$
I've been trying to figure it out, but to no avail. I believe it can be simplified to something where I can find the limit, but I cannot figure out how.
I am not even sure whether that helps when i split it into two series $\sum_{n=0}^\infty \frac{3^n}{(-7)^n}$ and $\sum_{n=0}^\infty \frac{4^n}{(-7)^n}$.
 A: Hints/Suggestions: Rewrite the summands in the form $(-3/7)^n + (-4/7)^n$ then evaluate by first evaluating $\sum_{n=0}^\infty (-3/7)^n$ and $\sum_{n=0}^\infty (-4/7)^n$ and then arguing that these are related to the series in question in the way one would expect.

Edit: In response to your edit, your intuition is right on the money; we would like to rewrite the series as
$$
\sum_{n=0}^\infty \frac{3^n+4^n}{(-7)^n} = \sum_{n=0}^\infty \left(-\frac{3}{7}\right)^n + \sum_{n=0}^\infty\left(-\frac{4}{7}\right)^n \qquad\qquad\qquad(\star)
$$
and then evaluate these two series using the geometric series identity $\sum_{n=0}^\infty x^n = \frac{1}{1-x}$ for $|x| < 1$. However, this isn't a precise argument yet. Specifically, it's not always true that
$$
    \sum_{n=0}^\infty (a_n + b_n) = \sum_{n=0}^\infty a_n + \sum_{n=0}^\infty b_n.
$$
A simple counterexample is to take $a_n = -1$ and $b_n = 1$ for all $n$; it's certainly true that $0 = \sum_{n=0}^\infty (-1+1)$, but the series $\sum_{n=0}^\infty \pm1$ both diverge, which means it makes no sense to say that $0 = \sum_{n=0}^\infty -1 + \sum_{n=0}^\infty 1$.
Fortunately, it is true that $\sum_{n=0}^\infty(a_n + b_n) = \sum_{n=0}^\infty a_n + \sum_{n=0}^\infty b_n$ if we know that $\sum_{n=0}^\infty a_n$ and $\sum_{n=0}^\infty b_n$ both converge. So here, for example, you can reach the conclusion that $(\star)$ is true but only after you've shown that $\sum_{n=0}^\infty(-3/7)^n$ and $\sum_{n=0}^\infty(-4/7)^n$ both converge.
A: This is just a sum of two geometric series and the absolute values of both rations are less then one. The only thing you need to use here is the standard trick that one uses for geometric series.
