1. **State the problem:** We want to find the real values of $x$ for which the series $\sum_{n=1}^\infty n^x$ converges. 2. **Recall the formula and rules:** The series is a p-series type but with general term $a_n = n^x$. For convergence of series $\sum n^p$, it converges if and only if $p < -1$. 3. **Apply the rule:** Here, $p = x$. So the series $\sum n^x$ converges if and only if $$x < -1$$ 4. **Explanation:** When $x < -1$, the terms $n^x = \frac{1}{n^{-x}}$ decrease fast enough to zero and the series converges. If $x \geq -1$, the terms do not decrease fast enough and the series diverges. **Final answer:** The series $\sum_{n=1}^\infty n^x$ converges for all real $x$ such that $$x < -1$$