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An explicit series requires us to present each term of the series clearly and distinctly. This detailed representation helps us understand exactly what we are summing up. In mathematics, when given a series with a summation notation like \( \sum_{m=1}^{4}(m^{2}+5) \), we replace the variable \( m \) with each integer value within the given range. This allows us to explicitly list the terms.For this series:- When \( m = 1 \), the calculation is \( 1^2 + 5 = 6 \)- When \( m = 2 \), the calculation is \( 2^2 + 5 = 9 \)- When \( m = 3 \), the calculation is \( 3^2 + 5 = 14 \)- When \( m = 4 \), the calculation is \( 4^2 + 5 = 21 \)We then express the series as \( 6 + 9 + 14 + 21 \). Explicitly listing these terms helps us visualize the problem and organize the numbers we will be adding together.

Evaluating the sum of a series involves adding together all the explicit terms that we have identified. This is a straightforward process once you have written out the series explicitly. Each term is added in sequence, which simplifies the complex expression into a single total value.In our example, the series \( 6 + 9 + 14 + 21 \) is evaluated by:

• First adding \( 6 \) and \( 9 \) to get \( 15 \)
• Then adding the result \( 15 \) with \( 14 \) to get \( 29 \)
• Lastly, adding \( 29 \) to \( 21 \) gives us the final sum of \( 50 \)

By showing each step in this manner, you can easily follow the process, ensuring accuracy and clarity in finding the sum of the series. This methodical addition ensures you understand how each part contributes to the complete sum.

Mathematical expressions are the foundation of any series problem. They consist of numbers, variables, and operators arranged to indicate a specific calculation or set of calculations. In this case, the expression \( m^2 + 5 \) combines arithmetic operations with variables.Understanding this expression involves:

• Recognizing that \( m^2 \) signifies \( m \times m \) (square of the variable \( m \))
• Including the constant value \( + 5 \) to each squared result
• Knowing that this needs to be computed for each value of \( m \) in the range specified by the summation symbol \( \sum_{m=1}^{4} \)

Once you've solved this type of mathematical expression for each individual term, it simplifies the task of series evaluation. Mastering these expressions empowers you to tackle any similar series problem effectively.