Problem 110 - Precalculus with Limits

Question

In Exercises 103-112, use sigma notation to write the sum. \( \dfrac{1}{1 \cdot 3} + \dfrac{1}{2 \cdot 4} + \dfrac{1}{3 \cdot 5} + \cdots + \dfrac{1}{10 \cdot 12} \)

Step-by-Step Solution

Answer

\( \sum_{n=1}^{10}\frac{1}{n(n+2)} \)

Step 1: Identify the Pattern

Look at the series and identify the pattern. For each term in the series, the numerator is incrementing by 1 starting from 1 and the denominator is also incrementing by 1 starting from 3.

Step 2: Write the General Term

Based on the pattern identified, write a general term that describes each term in the series. For this series, the general term is \(\frac{1}{n(n+2)}\) where n is the term number starting from 1.

Step 3: Write the Summation

Using the general term and the sigma notation, write the sum as a sigma notation. The sigma notation is written as \(\sum_{n=1}^{10}\frac{1}{n(n+2)}\), where 1 is the starting value, 10 is the ending value, and \(\frac{1}{n(n+2)}\) is the general term.

Key Concepts

Patterns in Sequences

Recognizing patterns in sequences is crucial when dealing with series and sums. In mathematics, a sequence is a collection of numbers arranged in a specific order, following a unique pattern. By observing the sequence and detecting its pattern, you can unlock the key to understanding its entire structure. In the given exercise, each term of the series is represented as fractions like \( \frac{1}{1 \cdot 3} \), \( \frac{1}{2 \cdot 4} \), and so forth. Here, we see a clear pattern:

• The numerator remains constant at 1.
• The sequential integers appear in both the multiplier of the denominator.
• The second number in the denominator is always two more than the first.

Understanding these patterns helps simplify the process of formulating a general term for the series. Patterns are like the skeleton of the sequence, providing the structure you need to organize and predict future terms.

General Term in Series

The general term in a series is a formula that encapsulates the pattern of the entire sequence. It allows you to express any term in the series without listing all the previous terms. In the exercise given, the terms follow a specific structure as identified from the pattern: for example, \( \frac{1}{n(n+2)} \). Here, "n" is a variable representing the position of a term in the sequence:

• When \( n = 1 \), the term is \( \frac{1}{1 \cdot 3} \).
• When \( n = 2 \), it becomes \( \frac{1}{2 \cdot 4} \).
• By continuing in this manner, you can compute any term in the series using the expression \( \frac{1}{n(n+2)} \).

Finding the general term aids in simplifying the task of expressing large series, making solutions efficient and manageable. It acts like a formulaic shortcut that can be used to explore further into properties or calculate specific parts of the series.

Notation in Mathematics

In mathematics, notation is a symbolic language that conveys complex ideas in a simplified form. When it comes to sequences and series, sigma notation is a powerful notation tool. Sigma notation, represented by the symbol \( \Sigma \), is used for expressing the sum of a sequence according to a specific pattern defined by its general term. In the problem at hand, the sequence is compactly represented as:\[\sum_{n=1}^{10}\frac{1}{n(n+2)}\]The sigma notation consists of several components:

• The \( \Sigma \) symbol itself, indicating that a sum is being taken.
• A general term \( \frac{1}{n(n+2)} \) that implies the pattern of terms being summed.
• A subscript \( n=1 \), which denotes the initial value of "n" in the series.
• A superscript \( 10 \), that marks the final value "n" should take in the summation.

This concise notation makes it much easier to understand and manipulate series and sums, especially when dealing with lengthy or complex sequences. Such mathematical shorthand is essential for communication and computation, facilitating more accessible problem-solving and concept exploration.