1. **Problem:** Find the sum of the series $$\frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \dots$$ 2. **Understanding the series:** Each term is of the form $$\frac{n}{n+1}$$ where $$n$$ starts from 1 and goes to infinity (or some finite number). 3. **Formula and simplification:** We can rewrite each term as: $$\frac{n}{n+1} = 1 - \frac{1}{n+1}$$ 4. **Sum of first $$N$$ terms:** $$S_N = \sum_{n=1}^N \frac{n}{n+1} = \sum_{n=1}^N \left(1 - \frac{1}{n+1}\right) = \sum_{n=1}^N 1 - \sum_{n=1}^N \frac{1}{n+1}$$ 5. **Evaluate sums:** $$\sum_{n=1}^N 1 = N$$ $$\sum_{n=1}^N \frac{1}{n+1} = \sum_{k=2}^{N+1} \frac{1}{k} = H_{N+1} - 1$$ where $$H_m$$ is the $$m$$-th harmonic number. 6. **Combine results:** $$S_N = N - (H_{N+1} - 1) = N + 1 - H_{N+1}$$ 7. **Interpretation:** The sum of the first $$N$$ terms is $$S_N = N + 1 - H_{N+1}$$. 8. **If infinite sum:** Since harmonic numbers grow without bound, the infinite sum diverges. **Final answer:** $$\boxed{S_N = N + 1 - H_{N+1}}$$