Questions: Use summation notation to write the series. 1 - 1/2 + 1/4 - 1/8 + ... - 1/2048

Solution Steps

To express the series in summation notation, observe the pattern of the terms. The series alternates signs and each term is a power of \(\frac{1}{2}\). The general term can be written as \((-1)^n \cdot \frac{1}{2^n}\). Determine the range of \(n\) by identifying the first and last terms.

The series is given by: \[ 1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \cdots - \frac{1}{2048} \]

The pattern of the series is alternating signs with each term being a power of \(\frac{1}{2}\). The general term can be expressed as: \[ a_n = (-1)^n \cdot \frac{1}{2^n} \]

The last term of the series is \(-\frac{1}{2048}\). We need to find \(n\) such that: \[ \frac{1}{2^n} = \frac{1}{2048} \]

Solving for \(n\): \[ 2^n = 2048 \] \[ n = \log_2(2048) = 11 \]

The series can be written in summation notation as: \[ \sum_{n=1}^{11} (-1)^n \cdot \frac{1}{2^n} \]

The sum of the series is: \[ \sum_{n=1}^{11} (-1)^n \cdot \frac{1}{2^n} = -0.3335 \]

Final Answer