Problem 74 - A Graphical Approach to College Algebra

Question

Use summation notation to write each series. Start the index at \(i=1\). $$\frac{1}{1+1}+\frac{2}{2+1}+\frac{3}{3+1}+\dots+\frac{25}{25+1}$$

Step-by-Step Solution

Answer

\( \sum_{i=1}^{25} \frac{i}{i+1} \)

Step 1: Identify the pattern

Examine the given series and identify the pattern of terms. Here, each term in the series can be expressed in the form \( \frac{i}{i+1} \), where \( i \) represents the position of the term in the sequence.

Step 2: Understand the range of the index

Determine the range of the index for the series. Since the series starts with the first term and ends with the 25th term, the variable \( i \) will go from 1 to 25.

Step 3: Write in summation notation

Combine the information from steps 1 and 2 to express the series using summation notation. This yields the following expression: \[ \sum_{i=1}^{25} \frac{i}{i+1} \].

Key Concepts

Series Representation

When working with series, one of the primary methods to represent them is through summation notation. This concise notation allows us to express long or infinite sequences in a compact form. Consider the series:

• Given the series: \( \frac{1}{1+1} + \frac{2}{2+1} + \frac{3}{3+1} + \dots + \frac{25}{25+1} \)
• We notice a repetitive pattern, which can be summarized efficiently using summation.
• By using the summation sign \( \sum \), we can express this entire series without listing each term individually, resulting in:\[ \sum_{i=1}^{25} \frac{i}{i+1} \]

This efficiently captures the entire sequence, saving space and offering clarity.

Index Range

The index range in summation notation is crucial because it defines the start and end of the series. It tells us which terms are included in the sequence. Here's how it works for our series:

• We start our index at 1, denoted by \( i=1 \). This means that the first term of our series is \( \frac{1}{1+1} \).
• The series continues until the index reaches 25, meaning our last term is \( \frac{25}{25+1} \).
• Therefore, the complete index range is from 1 to 25, which is written in our summation as \( \sum_{i=1}^{25} \).

Setting the correct index range ensures accuracy in representing and summing the series.

Pattern Identification

To effectively use summation notation, identifying the pattern in a series is essential. Recognizing the repeating structure helps in formulating the terms mathematically:

• Examine the given terms, \( \frac{1}{1+1}, \frac{2}{2+1}, \frac{3}{3+1}, \ldots \).
• Notice each numerator is simply \( i \), the position in the sequence, while each denominator is \( i+1 \).
• The pattern observed leads to each term being described as \( \frac{i}{i+1} \), involving the index \( i \).

Recognizing this pattern is key to correctly expressing the entire series in a mathematical form using summation notation.