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A series is an important concept in mathematics, representing the sum of a sequence of terms. In summation notation, a series is written using the sigma symbol (\( \sum \)), which signifies addition. The terms in a series are typically generated by an expression involving a variable, often represented by \( k \) or another letter. In our example, each term of the series is produced by the expression \(-2k\) as \( k \) varies over specified integer values.

To form a series, you evaluate the expression at each specified value of \( k \) and then add together the results. The original question asked us to evaluate \( \sum_{k=3}^{5} (-2k) \), meaning the expression \(-2k\) is calculated and summed for each integer from 3 to 5. This approach provided us with the series \(-6, -8, -10\), representing these evaluated expressions added together.

Understanding integer values is crucial for evaluating summation notation expressions. An integer is a whole number that can be positive, negative, or zero. In the context of a series, integer values typically represent the incrementing indices in summation notation.

When evaluating a series as described by \( \sum_{k=3}^{5} (-2k) \), we begin by recognizing that \( k \) takes on integer values from 3 through 5. Here, \( k \) is utilized as an index to plug into the expression, producing different terms of the series. By seamlessly substituting these integer values into the expression, you enable yourself to precisely generate the terms \(-2(3) = -6\), \(-2(4) = -8\), and \(-2(5) = -10\). This step is integral to fully understanding how integer values lead to accurately forming and calculating a mathematical series.

Expression evaluation is at the heart of working with summation notations and series. It involves substituting values into an expression and simplifying to arrive at a singular result. The primary goal here is to methodically evaluate each term involved in the summation.

For example, with the expression \(-2k\) and a summation from 3 to 5, you systematically replace \( k \) with each integer in this range. Starting with the smallest integer, 3, you calculate \(-2(3) = -6\). Next, for \( k = 4 \), evaluate \(-2(4) = -8\), and finally, substitute 5 to get \(-2(5) = -10\).

The careful evaluation of this expression at each integer value facilitates the construction of the series: \(-6, -8, -10\). Through these evaluations, the abstract properties of the expression become concrete values that make up the series, demonstrating the practical importance and application of expression evaluation in mathematical analysis.