1. **State the problem:** We want to find the sum of the infinite series $$\sum_{n=1}^\infty \frac{1}{2^n}$$ and verify that it equals 1. 2. **Formula used:** This is a geometric series with first term $$a = \frac{1}{2}$$ and common ratio $$r = \frac{1}{2}$$. 3. **Important rule:** The sum of an infinite geometric series where $$|r| < 1$$ is given by: $$S = \frac{a}{1-r}$$ 4. **Apply the formula:** $$S = \frac{\frac{1}{2}}{1 - \frac{1}{2}}$$ 5. **Simplify the denominator:** $$S = \frac{\frac{1}{2}}{\frac{1}{2}}$$ 6. **Cancel common factors:** $$S = \frac{\cancel{\frac{1}{2}}}{\cancel{\frac{1}{2}}} = 1$$ 7. **Conclusion:** The sum of the infinite series $$\sum_{n=1}^\infty \frac{1}{2^n}$$ is indeed 1.