1. **State the problem:** We want to find a formula for the partial sum $$S_n = \sum_{k=1}^n \left( \frac{1}{\sqrt{k+1}} - \frac{1}{\sqrt{k+2}} \right)$$ and then calculate the limit as $$n \to \infty$$. 2. **Write out the first few terms to see the telescoping pattern:** $$S_n = \left( \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{3}} \right) + \left( \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{4}} \right) + \cdots + \left( \frac{1}{\sqrt{n+1}} - \frac{1}{\sqrt{n+2}} \right)$$ 3. **Notice the telescoping effect:** Most terms cancel out, leaving only the first positive term and the last negative term: $$S_n = \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{n+2}}$$ 4. **Calculate the limit as $$n \to \infty$$:** Since $$\lim_{n \to \infty} \frac{1}{\sqrt{n+2}} = 0$$, we have $$\lim_{n \to \infty} S_n = \frac{1}{\sqrt{2}} - 0 = \frac{1}{\sqrt{2}}$$ **Final answers:** $$S_n = \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{n+2}}$$ $$\lim_{n \to \infty} S_n = \frac{1}{\sqrt{2}}$$