Problem 4 - Calculus Single Variable

Question

P In Exercises \(1-10\), write out the sum, and perform the addition. $$ \sum_{\ell=4}^{6} \ell /(\ell-3) $$

Step-by-Step Solution

Answer

The sum is 8.5.

Step 1: Understanding the Problem

We are given the notation \( \sum_{\ell=4}^{6} \frac{\ell}{\ell-3} \), which means we need to evaluate the expression \( \frac{\ell}{\ell-3} \) for each integer \( \ell \) from 4 to 6, and then add the results together.

Step 2: Evaluate Each Term

Substitute \( \ell = 4 \), \( \ell = 5 \), and \( \ell = 6 \) into the expression \( \frac{\ell}{\ell-3} \) and evaluate each:- For \( \ell = 4\), \( \frac{4}{4-3} = 4 \).- For \( \ell = 5\), \( \frac{5}{5-3} = 2.5 \).- For \( \ell = 6\), \( \frac{6}{6-3} = 2 \).

Step 3: Write Out the Sum

Now, write out the sum of the results from each term:\[ 4 + 2.5 + 2 \]

Step 4: Perform the Addition

Add the numbers together:\[ 4 + 2.5 + 2 = 8.5 \]

Step 5: Confirm the Result

Double-check each calculation to ensure correctness:- Each term is correctly evaluated.- The final sum \(8.5\) has been verified by adding the individual numbers.

Key Concepts

Summation

When you see the sigma notation, \( \sum \),it might look a little intimidating at first, but it's actually a pretty straightforward concept. The sigma symbol tells us that we are looking at a sum. This means we need to add up a sequence of numbers. In the case of our example exercise, the numbers we are adding together are from evaluating the expression \( \frac{\ell}{\ell-3} \)at specific integer values of \( \ell \).
This notation usually comes with three important components:

• The variable of summation (in our example, \( \ell \))
• The lower limit (the smallest integer value the variable will take, here it's 4)
• The upper limit (the largest integer value the variable will take, in our case, it's 6)

The sigma notation is a concise way to express the addition of terms that follow a specific pattern and saves us from writing out every term manually. It's really useful for expressing mathematical concepts that would otherwise require lots of repetitive writing.
Essentially, summation allows us to simplify expressing the process of adding sequences of numbers.

Rational Expressions

Rational expressions involve fractions where the numerator and/or the denominator are polynomials. In our exercise, \( \frac{\ell}{\ell-3} \) is a rational expression.
Understanding rational expressions is key because they appear frequently in many areas of mathematics. These types of expressions can sometimes seem complex, but they really boil down to simple polynomial fractions. The top part (numerator) and the bottom part (denominator) will both change as \( \ell \) changes.
One thing to be especially cautious about is the denominator. We need to make sure that it does not end up being zero as this would make the expression undefined. In our exercise, we carefully choose \( \ell \) values (4, 5, and 6) that do not result in a zero denominator, which keeps our calculations valid. By substituting the given values of \( \ell \), we simplify each fraction and can easily add them afterward.

Integer Evaluation

Integer evaluation involves substituting integer values into an expression, which simplifies the process of calculation. In our exercise, the integers 4, 5, and 6 are being substituted into the rational expression \( \frac{\ell}{\ell-3} \).
This step is crucial because it allows us to evaluate the expression at specific points, turning a general mathematical statement into specific, actionable terms. For each integer value:

• 4 gives us \( \frac{4}{1} \), which simplifies neatly to 4
• 5 translates to \( \frac{5}{2} \), or 2.5
• 6 simplifies to \( \frac{6}{3} \), which equals 2

By evaluating these, we perform the conversion from abstract notation into numerical values that can be summed up. This is an essential skill because it allows you to handle and resolve complex expressions step-by-step. Once all assigned integer substituting is complete, adding these together is straightforward, as seen in our final sum.