Problem 62 - Precalculus: Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The sum without sigma notation is \(0 + \sqrt{\frac{1}{3}} + \sqrt{\frac{1}{2}} + \sqrt{\frac{3}{5}}\).

Step 1: Understand Sigma Notation

The given sigma notation \(\sum_{j=1}^{4} \sqrt{\frac{j-1}{j+1}}\) represents the sum of a sequence of terms. Each term is \(\sqrt{\frac{j-1}{j+1}}\) where \(j\) is an integer that runs from 1 to 4.

Step 2: Expand the Sum

To write the sum without sigma notation, we need to expand it to individual terms. Substitute the integer values into the expression for each \(j\):- When \(j=1\), the term is \(\sqrt{\frac{1-1}{1+1}} = \sqrt{\frac{0}{2}} = 0\).- When \(j=2\), the term is \(\sqrt{\frac{2-1}{2+1}} = \sqrt{\frac{1}{3}}\).- When \(j=3\), the term is \(\sqrt{\frac{3-1}{3+1}} = \sqrt{\frac{2}{4}} = \sqrt{\frac{1}{2}}\).- When \(j=4\), the term is \(\sqrt{\frac{4-1}{4+1}} = \sqrt{\frac{3}{5}}\).

Step 3: Write the Sum without Sigma

The expression for the sum without using sigma notation is the addition of the terms obtained from substituting \(j\) values:\[0 + \sqrt{\frac{1}{3}} + \sqrt{\frac{1}{2}} + \sqrt{\frac{3}{5}}\].

Key Concepts

Sequence of Terms

When we talk about a "sequence of terms," we are referring to a set of numbers arranged in a specific order. Each number in this set is called a term. Understanding sequences is crucial when working with sigma notation in mathematics. Sigma notation is used to represent the sum of such sequences.
The given exercise involves a particular sequence where each term is defined by the expression \( \sqrt{\frac{j-1}{j+1}} \). Here, \( j \) takes integer values from 1 through 4.
In this context:

• For \( j = 1 \), the term is \( \sqrt{\frac{1-1}{1+1}} = 0 \).
• For \( j = 2 \), the term is \( \sqrt{\frac{2-1}{2+1}} = \sqrt{\frac{1}{3}} \).
• For \( j = 3 \), the term is \( \sqrt{\frac{3-1}{3+1}} = \sqrt{\frac{1}{2}} \).
• For \( j = 4 \), the term is \( \sqrt{\frac{4-1}{4+1}} = \sqrt{\frac{3}{5}} \).

This sequence guides us in determining each term that is included in the summation.

Summation Expansion

Summation expansion is the process of expressing a sigma notation as an explicit sum of its individual terms.
Sigma notation is compact and shows the form and limit of the summation but doesn't specify each individual term's value directly.
In the exercise, expanding the summation \( \sum_{j=1}^{4} \sqrt{\frac{j-1}{j+1}} \) means writing down each term from the sequence:

• The first term: \( 0 \)
• The second term: \( \sqrt{\frac{1}{3}} \)
• The third term: \( \sqrt{\frac{1}{2}} \)
• The fourth term: \( \sqrt{\frac{3}{5}} \)

By substituting each specific value of \( j \) into the expression \( \sqrt{\frac{j-1}{j+1}} \), we translate the sigma notation's compact form into a detailed summation:
\[ 0 + \sqrt{\frac{1}{3}} + \sqrt{\frac{1}{2}} + \sqrt{\frac{3}{5}} \].
This method allows us to understand the contribution of each term to the overall sum.

Mathematical Expression

A mathematical expression involves numbers, operations, and/or variables arranged in a meaningful way. In the context of this exercise, \( \sqrt{\frac{j-1}{j+1}} \) is the mathematical expression.
This expression is applied to each integer \( j \) within the specified range, giving us the sequence of terms. Let's detail it:

• The numerator \( j-1 \) reduces the current integer by one.
• The denominator \( j+1 \) increases the current integer by one.
• The square root \( \sqrt{} \) encapsulates the whole fraction \( \frac{j-1}{j+1} \).

Using this expression, we calculate each term needed in the sum. It's essential to thoroughly understand these components because altering any could result in a completely different sequence.
This exemplifies how mathematical expressions form the backbone of deriving insights in mathematics, whether it's summing terms or modeling complex phenomena.