1. **Problem Statement:** Compute the Riemann sum $$\sum_{i=1}^n \left(1 + \frac{2i}{n}\right) \frac{1}{n}$$ and find its limit as $$n \to \infty$$. 2. **Understanding the Riemann Sum:** This sum approximates the integral of a function over an interval by dividing it into $$n$$ subintervals. Here, the function is $$f(x) = 1 + 2x$$ evaluated at points $$x_i = \frac{i}{n}$$, and the width of each subinterval is $$\Delta x = \frac{1}{n}$$. 3. **Rewrite the sum:** $$\sum_{i=1}^n \left(1 + \frac{2i}{n}\right) \frac{1}{n} = \sum_{i=1}^n \frac{1}{n} + \sum_{i=1}^n \frac{2i}{n^2} = \sum_{i=1}^n \frac{1}{n} + 2 \sum_{i=1}^n \frac{i}{n^2}$$ 4. **Evaluate each sum separately:** - $$\sum_{i=1}^n \frac{1}{n} = n \times \frac{1}{n} = 1$$ - $$\sum_{i=1}^n i = \frac{n(n+1)}{2}$$, so $$2 \sum_{i=1}^n \frac{i}{n^2} = 2 \times \frac{1}{n^2} \times \frac{n(n+1)}{2} = \frac{n+1}{n}$$ 5. **Combine results:** $$1 + \frac{n+1}{n} = 1 + 1 + \frac{1}{n} = 2 + \frac{1}{n}$$ 6. **Find the limit as $$n \to \infty$$:** $$\lim_{n \to \infty} \left(2 + \frac{1}{n}\right) = 2 + 0 = 2$$ **Final answer:** The limit of the Riemann sum is $$\boxed{2}$$.