Problem 122 - Calculus for Biology and Medicine

Question

Write each sum in expanded form. $$ \sum_{k=1}^{3} \frac{k^{2}}{k^{2}+1} $$

Step-by-Step Solution

Answer

The expanded form is \( \frac{1}{2} + \frac{4}{5} + \frac{9}{10} \).

Step 1: Understand the Summation Notation

The given summation notation is \( \sum_{k=1}^{3} \frac{k^{2}}{k^{2}+1} \). This notation means we need to evaluate the fraction \( \frac{k^{2}}{k^{2}+1} \) starting from \( k = 1 \) up to \( k = 3 \) and then sum these values together.

Step 2: Substitute for \(k=1\)

Substitute \( k = 1 \) into the expression \( \frac{k^{2}}{k^{2}+1} \). This results in: \[ \frac{1^{2}}{1^{2}+1} = \frac{1}{2} \]

Step 3: Substitute for \(k=2\)

Substitute \( k = 2 \) into the expression \( \frac{k^{2}}{k^{2}+1} \). This results in: \[ \frac{2^{2}}{2^{2}+1} = \frac{4}{5} \]

Step 4: Substitute for \(k=3\)

Substitute \( k = 3 \) into the expression \( \frac{k^{2}}{k^{2}+1} \). This results in: \[ \frac{3^{2}}{3^{2}+1} = \frac{9}{10} \]

Step 5: Write the Expanded Form

Now, combine the results from Steps 2, 3, and 4 to write the problem in expanded form. The expanded form is: \[ \frac{1}{2} + \frac{4}{5} + \frac{9}{10} \]

Key Concepts

Expanded Form

When we talk about the expanded form in the context of summation notation, we're referring to the process of taking a generalized formula and expressing it as a sum of individual values. Think of it as unfolding a neat package to see each item inside. The expanded form makes the math operation clearly visible by representing each term that will be summed.

In our exercise, we start with the summation notation \( \sum_{k=1}^{3} \frac{k^{2}}{k^{2}+1} \). This compact expression indicates that we will substitute each integer value from 1 to 3 into the fraction \( \frac{k^{2}}{k^{2}+1} \) and add up the results. By moving through the values of \( k \), the expanded form becomes:

• Substitute \( k=1 \): \( \frac{1}{2} \)
• Substitute \( k=2 \): \( \frac{4}{5} \)
• Substitute \( k=3 \): \( \frac{9}{10} \)

Thus, the expanded form is simply adding these fractions together to form \( \frac{1}{2} + \frac{4}{5} + \frac{9}{10} \). This format not only helps in understanding the progression but also aids in computational steps that follow.

Fraction Evaluation

Fraction evaluation involves the process of calculating the numeric value of fractional expressions by substituting specific values into them. In the given example, this process is an essential step to transform the summation expression into its expanded form.

Each value of \( k \) is substituted into the expression \( \frac{k^{2}}{k^{2}+1} \), resulting in different fractions based on the integer chosen. Here’s how the fractions are evaluated:

• For \( k=1 \), compute \( \frac{1^{2}}{1^{2}+1} = \frac{1}{2} \)
• For \( k=2 \), compute \( \frac{2^{2}}{2^{2}+1} = \frac{4}{5} \)
• For \( k=3 \), compute \( \frac{3^{2}}{3^{2}+1} = \frac{9}{10} \)

Each fraction is derived by squaring the value of \( k \) and performing basic algebraic operations. It's important to carefully follow arithmetic rules to ensure the accuracy of each evaluated fraction. Mastering fraction evaluation in such expressions is key to simplifying larger, more complex problems.

Mathematical Expression Substitution

Mathematical expression substitution is a technique wherein specific numbers replace variables within an expression to find concrete values. This approach is fundamental for transforming abstract mathematical notations into practical numerical solutions.

In our case, the expression \( \frac{k^{2}}{k^{2}+1} \) is a general formula representing various values depending on \( k \). To evaluate it, each designated value of \( k \) from the summation range is substituted to calculate the corresponding fractional result.

Here's how substitution procedures look for this exercise:

• Insert \( k=1 \) into \( \frac{k^{2}}{k^{2}+1} \) yielding \( \frac{1}{2} \)
• Insert \( k=2 \) into \( \frac{k^{2}}{k^{2}+1} \) yielding \( \frac{4}{5} \)
• Insert \( k=3 \) into \( \frac{k^{2}}{k^{2}+1} \) yielding \( \frac{9}{10} \)

Substitution helps decipher what each term in the summation actually represents. By computing these substitutions for all required values, summation notation transitions from abstract to palpable numerical sums, leading to a clearer understanding of underlying patterns or series.