Problem 24 In $15-26,$ write each series ... [FREE SOLUTION]

Chapter 6: Problem 24

In $15-26,$ write each series in sigma notation. $$ \frac{1}{1 \times 2}+\frac{1}{2 \times 3}+\frac{1}{3 \times 4}+\frac{1}{4 \times 5}+\frac{1}{5 \times 6}+\frac{1}{6 \times 7} $$

Short Answer

Expert verified

$ \sum_{n=1}^{6} \frac{1}{n(n+1)} $

Step by step solution

Analyze the Series Structure

First, identify the pattern in the series. Observe that each term is of the form $ \frac{1}{n(n+1)} $ where $ n $ is an integer that starts from 1 and increases by 1 for each subsequent term.

Determine the Range of Summation Index

Determine the smallest and largest values of $ n $ used in the series. The given series starts at $ \frac{1}{1 \times 2} $ which corresponds to $ n = 1 $, and ends at $ \frac{1}{6 \times 7} $ corresponding to $ n = 6 $. Thus the series runs from $ n = 1 $ to $ n = 6 $.

Write in Sigma Notation

Using the general form of the terms and the identified range of $ n $, express the series in sigma notation: \[ \sum_{n=1}^{6} \frac{1}{n(n+1)} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Series

An algebraic series is essentially a sequence of numbers where each term is determined using a specific algebraic pattern or formula. These series can have patterns based on arithmetic operations like addition, multiplication, division, or exponentiation.

In our exercise, the algebraic series is generated by a pattern involving division, where each term is expressed as $ \frac{1}{n(n+1)} $. Here, each term is calculated by first identifying $ n $, then using it in the term's formula.

The sequence starts with $ n = 1 $, creating the term $ \frac{1}{1 \times 2} $.
It continues, with each next term using the subsequent integer value of $ n $.
This pattern follows logically, ensuring a smooth transition between consecutive terms of the series.

Understanding the pattern is crucial to writing the series as a summation and helps in identifying how each term is structured.

Sigma Notation

Sigma notation is a way of writing a long series of terms in a compact and concise form. It uses the Greek letter $ \Sigma $ (sigma) to indicate summation, making it a powerful tool for handling series in mathematics.

In the provided exercise, the series is expressed using sigma notation as $ \sum_{n=1}^{6} \frac{1}{n(n+1)} $, which denotes the sum of the terms calculated from $ n = 1 $ to $ n = 6 $.

The expression $ \frac{1}{n(n+1)} $ inside the sigma symbol represents the general form of each term in the series.
The numbers below and above the sigma specify the starting and ending values of the summation index $ n $.
This notation simplifies the representation of the series and aids in performing algebraic operations more efficiently.

By highlighting the series鈥� structure through sigma notation, complex calculations can be better understood and communicated.

Summation Index

The summation index, often denoted as $ n $, is a placeholder that represents each term's position within a series in sigma notation. This index acts as a variable that varies over a specified range, defining every term in the series.

For our series, the summation index $ n $ starts at 1 and goes up to 6. This dictates that the series includes terms for all integer values of $ n $ between 1 and 6.

The start of the summation index, $ n=1 $, corresponds to the term $ \frac{1}{1 \times 2} $.
Each subsequent term increases the summation index by 1, continuing the pattern.
Finishing at $ n=6 $ with the term $ \frac{1}{6 \times 7} $, the series covers all terms from 1 through 6.

Understanding the role of the summation index is crucial, as it sets the limits and pattern for the series within the sigma notation framework.

Mathematical Series Structure

The mathematical series structure refers to the ordered arrangement of numbers where each follows a rule or formula. The structure is fundamental to comprehending the essence and progression of a series.

In the series given in the exercise, each term's structure adheres to the formula $ \frac{1}{n(n+1)} $. This coherent pattern ensures that each part of the series connects seamlessly.

Each term is clearly defined by the formula, involving multiplication and division, crucial for consistency.
The sequence naturally progresses by adjusting the index $ n $.
Keeping track of the structure helps in identifying how each part fits into the series as a whole.

By recognizing and understanding the series鈥� structure, we can grasp how it forms an organized and logical mathematical entity, facilitating better analysis and manipulation in further mathematical operations.

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91影视

In \(15-26,\) write each series in sigma notation. $$ \frac{1}{1 \times 2}+\frac{1}{2 \times 3}+\frac{1}{3 \times 4}+\frac{1}{4 \times 5}+\frac{1}{5 \times 6}+\frac{1}{6 \times 7} $$

Short Answer

Step by step solution

Analyze the Series Structure

Determine the Range of Summation Index

Write in Sigma Notation

Key Concepts

Algebraic Series

Sigma Notation

Summation Index

Mathematical Series Structure

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