1. **Stating the problem:** We want to express the series $$1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \frac{1}{16} - \frac{1}{32} + \frac{1}{64}$$ as a summation notation and identify which given option is correct. 2. **Observing the pattern:** The terms alternate in sign and the denominators are powers of 2. The first term is $1 = \frac{1}{2^0}$, the second term is $-\frac{1}{2^1}$, the third is $\frac{1}{2^2}$, and so on. 3. **General term:** The $n$-th term can be written as $$(-1)^n \cdot \frac{1}{2^n} = (-1)^n 2^{-n}$$ where $n$ starts at 0. 4. **Number of terms:** There are 7 terms, so $n$ goes from 0 to 6. 5. **Summation notation:** $$\sum_{n=0}^6 (-1)^n 2^{-n}$$ 6. **Checking options:** - Option A: $\sum_{n=1}^6 (-1)^n 2^{-2}$ is incorrect because the power of 2 is constant $-2$ and the index starts at 1. - Option B: $\sum_{n=0}^6 (-1)^n 2^{-n}$ matches our derived formula. - Option C: $\sum_{n=1}^6 (-1)^n 2^n$ is incorrect because powers of 2 are positive exponents and index starts at 1. - Option D: $\sum_{n=0}^6 (-1)^n 2^n$ is incorrect because powers of 2 are positive exponents. **Final answer:** Option B is correct. $$\boxed{\sum_{n=0}^6 (-1)^n 2^{-n}}$$