Problem 16 - Calculus Single Variable

Question

In Exercises \(11-16,\) use summation notation to express the sum. $$ 2 / 5+3 / 7+4 / 9+5 / 11 $$

Step-by-Step Solution

Answer

The sum is expressed as \( \sum_{n=1}^{4} \frac{n+1}{2n+3} \).

Step 1: Identify the Sequence Pattern

First, identify the pattern in both the numerators and denominators of the fractions given in the sequence: \( \frac{2}{5}, \frac{3}{7}, \frac{4}{9}, \frac{5}{11} \). Notice that the numerators increase by 1 starting from 2, and the denominators increase by 2 starting from 5.

Step 2: Define the Terms of the Sequence

Express each term in a general form. Let the term be \( \frac{n+1}{2n+3} \). Here, \( n \) starts at 1 (to give the first term \( \frac{2}{5} \)) and ends at 4 (to give the last term \( \frac{5}{11} \)).

Step 3: Set Up the Summation Notation

Using summation notation, express the series as: \[ \sum_{n=1}^{4} \frac{n+1}{2n+3} \].This represents the sum of the sequence from \( n = 1 \) to \( n = 4 \) using the expression \( \frac{n+1}{2n+3} \).

Key Concepts

Sequence Pattern

Understanding sequence patterns is like finding a rhythm in a song. A sequence in mathematics is essentially a set of numbers arranged in a particular order. These numbers follow a specific rule or pattern, which helps us anticipate what comes next. In our exercise, we're looking at fractions: \( \frac{2}{5}, \frac{3}{7}, \frac{4}{9}, \frac{5}{11} \). As you can observe, there is a clear pattern in both the numerators and denominators of these fractions.

• The numerators (2, 3, 4, 5) increase by 1.
• The denominators (5, 7, 9, 11) grow by 2.

By recognizing these patterns, we can better understand how sequences are structured. This is crucial for solving sequence-related problems.

Numerators and Denominators

In fractions, the numerator is the top number, and the denominator is the bottom number. Each provides unique information about part-whole relationships.

In our sequence of fractions, the numerators increase one step at a time. For instance, it starts at 2 and goes up to 5. This gradual change gives each fraction a sense of progression.

• At \(n = 1\), numerator = 2.
• At \(n = 2\), numerator = 3.
• At \(n = 3\), numerator = 4.
• At \(n = 4\), numerator = 5.

Meanwhile, the denominators rise by 2 each time. For instance, the first denominator is 5 and the last is 11.

• Starting denominator (at \(n = 1\)) = 5.
• Next (at \(n = 2\)) = 7.
• Then (at \(n = 3\)) = 9.
• Ending (at \(n = 4\)) = 11.

Recognizing how each part of the fraction changes is key to understanding the mathematics behind the sequence.

General Term

The general term of a sequence is like its DNA—it defines every element. Once identified, it allows us to compute any term in the sequence given its position.

For the sequence in question, we can express each term as \( \frac{n+1}{2n+3} \). This formula lets us pinpoint each term based on \(n\), where \(n\) represents the position of the term in the sequence.

• For \(n = 1\): \(\frac{1+1}{2 \cdot 1 + 3} = \frac{2}{5}\).
• For \(n = 2\): \(\frac{2+1}{2 \cdot 2 + 3} = \frac{3}{7}\).
• For \(n = 3\): \(\frac{3+1}{2 \cdot 3 + 3} = \frac{4}{9}\).
• For \(n = 4\): \(\frac{4+1}{2 \cdot 4 + 3} = \frac{5}{11}\).

Using the general form, we can easily check if our terms are correct and figure out additional terms if needed. This deep understanding of sequences can simplify complex problems.

Series

A series in mathematics is like a marathon; it stretches over a sequence but focuses on adding them together. Specifically, a series represents the sum of terms in a sequence.

In our exercise, the series is represented using summation notation: \( \sum_{n=1}^{4} \frac{n+1}{2n+3} \).

• This notation means we're adding up each term from \(n=1\) to \(n=4\).
• This adds up precisely the fractions \(\frac{2}{5}\), \(\frac{3}{7}\), \(\frac{4}{9}\), and \(\frac{5}{11}\).

Converting sequences into series through summation helps answer questions about total quantities and means we can find the collective sum easily. The use of the sigma \(\Sigma\) symbol is an elegant way to handle long sums without literally writing every individual term. This notation not only simplifies our calculations but also helps create a standardized, streamlined way to approach mathematics.