Summation of a Geometric Series using Sigma Notation

Using the summation notation $\Sigma$, calculate the sum of the geometric series from $k=2$ to $k=7$ with the geometric rule $a_k = \frac{1}{2}^k \times 2$.

This problem involves calculating the sum of a geometric series using summation notation, one of the fundamental notations in calculus for representing series and sequences. Here, summation notation (often represented by the Greek letter Sigma) is used to compactly express the sum of a series, in this case, a geometric series. A geometric series is characterized by each term being multiplied by a constant factor from one term to the next. In this problem, you are tasked with evaluating a specific range of terms in the series from the second term to the seventh term, with a given formula for each term in the series. Understanding the general form and behavior of geometric series, and how to use summation notation effectively, is key to solving this problem.

Additionally, note that the series described has a common ratio that affects the behavior and convergence of the series, as the terms decrease rapidly. Handling such series often involves recognizing patterns and understanding how to simplify expressions under the summation notation. Solutions involve algebraic manipulation and insights into the convergence and sum formula of geometric series especially as limits are approached or bounds are defined, which are common exercises in the study of sequences and series in calculus.

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