1. **State the problem:** Find the limit as $n$ approaches infinity of the sequence $$a_n = 1 + \sqrt{2} + \sqrt{3} + \cdots + \sqrt{n}.$$\n\n2. **Understanding the problem:** We want to evaluate $$\lim_{n \to \infty} a_n = \lim_{n \to \infty} \sum_{k=1}^n \sqrt{k}.$$\n\n3. **Approach:** Since the sum involves square roots, we can approximate the sum by an integral to understand its growth behavior. The function $f(x) = \sqrt{x}$ is increasing and positive for $x \geq 1$.\n\n4. **Integral approximation:** The sum $$\sum_{k=1}^n \sqrt{k}$$ can be approximated by the integral $$\int_1^n \sqrt{x} \, dx.$$\n\n5. **Calculate the integral:**\n$$\int_1^n \sqrt{x} \, dx = \int_1^n x^{1/2} \, dx = \left[ \frac{2}{3} x^{3/2} \right]_1^n = \frac{2}{3} (n^{3/2} - 1).$$\n\n6. **Interpretation:** As $n \to \infty$, $n^{3/2}$ dominates, so the sum grows approximately like $\frac{2}{3} n^{3/2}$.\n\n7. **Conclusion:** The sum $$1 + \sqrt{2} + \sqrt{3} + \cdots + \sqrt{n}$$ diverges to infinity as $n$ approaches infinity. Therefore, $$\lim_{n \to \infty} \sum_{k=1}^n \sqrt{k} = \infty.$$