# How to show if $\sum_{n=1}^\infty 7X_{n+1} + X_n$ converges and $X_n \to 0$ when $n \to \infty$, then $\sum_{n=1}^\infty X_n$ converges?

Note that $$X_n$$ is just a sequence of real numbers.

I'm a bit out of ideas because I'm used to having the extra hypothesis of $$X_n$$ being of positive terms. Any hints or suggestions will be greatly appreciated!

Let $$L:=\sum_{n=1}^\infty (7X_{n+1} + X_n)$$ and $$S_N=\sum_{n=1}^N X_n$$. Note that $$\sum_{n=1}^N( 7X_{n+1} + X_n) = 8S_N + 7X_{N+1}-7X_1$$ thus $$\lim_{N\to \infty} (8S_N + 7X_{N+1}-7X_1) = L$$. From the hypothesis $$\lim_N X_N=0$$, we can infer that $$S_N$$ converges and $$\lim_{N\to \infty} S_N = \frac 18(7X_1+L)$$.
Consider $$S_k := \sum_{n=1}^k x_n$$. Then your condition reads: \begin{align} (i) \quad & 7(S_{k+1}-S_1) + S_k \to L \quad \text{as } k \to \infty; \\ (ii) \quad & S_{k+1}-S_k \to 0 \quad \text{as } k \to \infty. \end{align} Note that $$(i)$$ is equivalent to saying that $$7S_{k+1} + S_k$$ converges, since $$S_1$$ is a fixed real number.
Now note that $$7S_{k+1}+S_k = 8S_{k+1}-(S_{k+1}-S_k)$$ converges. Can you use conition $$(ii)$$ to deduce the desired result?
$$\sum _{n=1}^\infty 7X_{n+1}+X_n=X_1+8\sum _{n=2}^\infty X_n$$ The second hypothesis is unneccessary. Indeed, it is implied by the first.