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Summation is the process of adding a sequence of numbers or expressions. It is usually symbolized by the Greek letter sigma ( \(\sum\) ). Summation can be thought of as the repeated addition of a sequence of numbers. The sequence of numbers is defined by a function, and the summation takes the form of adding these outputs together.
For example, in the exercise, the summation notation \(\sum_{i=1}^{4} g\(x_{i}\)\) tells us to take the values of the function g at different points \(\left( x_{1}, x_{2}, x_{3}, x_{4} \right)\) and sum them up.
This is particularly useful when dealing with large sequences as it offers a compact way to represent such additions.

Notation refers to the method of writing out mathematical concepts in symbols and numbers. In the case of summation, summation notation involves the sigma symbol ( \(\sum\) ) followed by an expression that indicates the terms to be added and the range of indices over which to sum.

For example, the notation \(\sum_{i=1}^{4} g\(x_{i}\)\) can be broken down into:

• The sigma symbol specifies that addition will occur.
• i = 1 to 4 denotes the range of the summation, meaning that i will take values from 1 through 4.
• g(x_{i}) indicates the formula whose terms are to be summed.

Understanding summation notation can simplify reading and writing complex mathematical relationships involving sequences.

An explicit sum means writing out the entire sum step-by-step, without any shorthand notation. This helps in understanding and evaluating the individual terms that make up the summation.
In the given exercise, the summation notation \(\sum_{i=1}^{4} g\(x_{i}\)\) can be rewritten explicitly as:

\( g\(x_{1}\) + g\(x_{2}\) + g\(x_{3}\) + g\(x_{4}\)\)
Breaking up summation notations like this can be quite helpful in simplifying complex expressions in mathematics. It lets you see every term involved in the sum, making it easier to perform calculations and spot potential errors.

To thoroughly understand the exercise, it's important to follow a step-by-step approach. Here's how to do it:

1. **Identify the range of the summation**: The problem states \(\sum_{i=1}^{4} g\(x_{i}\)\). This tells us that the summation index 'i' ranges from 1 to 4.
2. **Write each term individually**: Next, we need to express every term in the sequence. For i = 1, 2, 3, and 4, write out the function evaluations: \(\left[g(x_{1}), g(x_{2}), g(x_{3}), g(x_{4})\right]\).
3. **Combine the terms**: Finally, sum up these individual terms to express the total explicit sum:

\( g\(x_{1}\) + g\(x_{2}\) + g\(x_{3}\) + g\(x_{4}\) \).
Following these steps ensures a clear and precise understanding of how the summation breaks down.