Problem 56 - Precalculus Mathematics for Calculus

Question

Write the sum without using sigma notation. $$\sum_{i=0}^{4} \frac{2 i-1}{2 i+1}$$

Step-by-Step Solution

Answer

The sum is \(-1 + \frac{1}{3} + \frac{3}{5} + \frac{5}{7} + \frac{7}{9}.\)

Step 1: Understand the Sigma Notation

The given expression uses sigma (∑) notation, which represents the sum of a sequence. The expression \( \sum_{i=0}^{4} \frac{2i-1}{2i+1} \) indicates that we need to compute the sum of the terms \( \frac{2i-1}{2i+1} \) starting with \( i = 0 \) up to \( i = 4 \).

Step 2: Compute Individual Terms

Compute the terms for each value of \( i \) from 0 to 4:- For \( i = 0 \), term = \( \frac{2(0)-1}{2(0)+1} = \frac{-1}{1} = -1 \).- For \( i = 1 \), term = \( \frac{2(1)-1}{2(1)+1} = \frac{1}{3} \).- For \( i = 2 \), term = \( \frac{2(2)-1}{2(2)+1} = \frac{3}{5} \).- For \( i = 3 \), term = \( \frac{2(3)-1}{2(3)+1} = \frac{5}{7} \).- For \( i = 4 \), term = \( \frac{2(4)-1}{2(4)+1} = \frac{7}{9} \).

Step 3: Write the Sum

Now that you have calculated each individual term, write down the sum of these terms: \(-1 + \frac{1}{3} + \frac{3}{5} + \frac{5}{7} + \frac{7}{9}.\) This represents the sum without using sigma notation.

Key Concepts

Summation without Sigma Notation

Summation is the process of adding a sequence of numbers together to achieve a total. In mathematical expressions, a common way to denote summation is through sigma notation. The Greek letter "Σ" symbolizes the starting point for summing a sequence of numbers. For our exercise, sigma notation provides a concise way to represent multiple additions. The expression \( \sum_{i=0}^{4} \frac{2i-1}{2i+1} \) denotes adding terms like \( \frac{2(0)-1}{2(0)+1} \) through \( \frac{2(4)-1}{2(4)+1} \) together:

• The lower index \(i = 0\) is where the sequence begins.
• The upper index \(i = 4\) indicates where it ends.

Therefore, when the sigma notation is expanded, it results in individually calculated terms added up: \(-1 + \frac{1}{3} + \frac{3}{5} + \frac{5}{7} + \frac{7}{9}\). This step allows us to verify each part and make sure the whole sequence works accurately.

Understanding Sequences in Mathematics

A sequence in math is a particular order of numbers that follow a specific rule or function. Sequences can be finite, with a clear end, or infinite, going on forever.In our example, the sequence starts at \(i=0\) and ends at \(i=4\). The terms of the sequence are generated using the formula: \(\frac{2i-1}{2i+1}\). Each term differs by using consecutive integers in the place of \(i\) and calculating the result:

• Substitute the values \(i = 0, 1, 2, 3, 4\).
• Plug these into the formula \(\frac{2i-1}{2i+1}\).
• Compute each resulting fraction.

These calculated fractions comprise the elements of our sequence, which together, when summed, give the total we need.

Precalculus and its Importance

Precalculus acts as a bridge from basic algebra and geometry to more advanced topics, such as calculus. It's a stepping stone that prepares students by building foundational skills. Topics within precalculus often include sequences, series, functions, and trigonometry. Here, sequences and summation form a crucial part. You gain the groundwork for understanding calculus concepts like limits and the behavior of functions by mastering them:

• Insight into calculating sums efficiently with sigma notation.
• Building familiarity with sequences for predicting future behaviors of functions.

Lessons like this one serve to develop critical thinking and problem-solving techniques required for more abstract calculus challenges, creating a smooth transition into higher-level math.