Problem 128 - Calculus for Biology and Medicine

Question

Write each sum in sigma notation. \(\frac{3}{5}+\frac{4}{6}+\frac{5}{7}+\frac{6}{8}+\frac{7}{9}\)

Step-by-Step Solution

Answer

\( \sum_{k=1}^{5} \frac{k+2}{k+4} \)

Step 1: Understand the Pattern

The given sum is \( \frac{3}{5}+\frac{4}{6}+\frac{5}{7}+\frac{6}{8}+\frac{7}{9} \). Each term in the sum consists of a numerator and a denominator that increase by 1 with each subsequent term.

Step 2: Identify the General Term

From the pattern, the numerator starts at 3 and increases by 1 in each term, while the denominator starts at 5 and increases by 1 in each term. The k-th term can thus be expressed as \( \frac{k+2}{k+4} \).

Step 3: Determine the Index Range

Determine the range of k. The first term corresponds to \( k+2=3 \), so \( k=1 \). The last term corresponds to \( k+4=9 \), so \( k=5 \). Hence, k ranges from 1 to 5.

Step 4: Write in Sigma Notation

Now, we can write the sum in sigma notation using the general term and the range for k: \[\sum_{k=1}^{5} \frac{k+2}{k+4}\]

Key Concepts

Series

In mathematics, a series is a summation of a sequence of numbers. Each number in the sequence is defined as a term. When we talk about a series, we mean adding up all these terms to get a total. For instance, if you have a sequence like: 2, 4, 6, 8, the related series is the sum: 2 + 4 + 6 + 8.
In our example from the exercise, the series involves the addition of fractions: \(\frac{3}{5}, \frac{4}{6}, \frac{5}{7}, \frac{6}{8}\), and \(\frac{7}{9}\). Here, each fraction is a term, and together they make a complete series.
Series help in finding the total of elements that change according to a specific pattern. Often, we describe such a series using sigma notation to concisely express this sum.

Index Range

The index range is crucial in understanding series expressed in sigma notation. It defines the starting and ending points of the sequence terms we sum up. In sigma notation, the variable (often \(k\)) under the sigma symbol (\(\Sigma\)) tells us the index range.
In the series \(\frac{3}{5} + \frac{4}{6} + \frac{5}{7} + \frac{6}{8} + \frac{7}{9}\), we determined that \(k\) starts at 1 and ends at 5. This means that the terms we sum begin from \(k=1\) and continue through to \(k=5\).
Understanding the index range helps in knowing how many terms are in the sum and which values to use in the general term formula to generate all terms to be added.

General Term

Identifying a general term is an essential step in writing a series in sigma notation. It represents each term of the sequence using an expression with the index variable \(k\). The general term provides a formula so that any term in the sequence can be calculated by plugging in values from the index range.
In our example, the general term is \(\frac{k+2}{k+4}\). This formula was derived by observing the pattern in the given sequence of fractions.
- The numerator starts at 3 and increases by 1 with each term, simplifying to \(k+2\). - Similarly, the denominator starts at 5 and also increases by 1 with each term, simplifying to \(k+4\).
By using the general term, you can easily generate all necessary terms within the specified index range.