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Chapter 12: Problem 11

Express in sigma notation. $$1+3+5+7 \cdots \cdots+21$$

Short Answer

Expert verified
The given arithmetic series can be expressed in sigma notation as \(\sum_{n=1}^{11} (2n - 1)\).

Step by step solution

01

Identify the pattern or formula for generating odd numbers

Arithmetic series consists of a sequence of numbers where the common difference between consecutive terms is constant. In our case, the difference between each consecutive odd number is 2. Therefore, the formula for generating the nth odd number can be given as: \(a_n = 1 + (n - 1) * 2\), or simply, \(a_n = 2n - 1\). Now that we have the formula for generating odd numbers, let's determine the number of terms in the series.
02

Determine the number of terms in the series

Using the formula for the nth odd number, we replace \(a_n\) with 21 (since it is the last term in the series): \(21 = 2n - 1\) Now, solve for n: \(21 + 1 = 2n\) \(\frac{22}{2} = n\) Therefore, n = 11. There are 11 terms in the series.
03

Express the series in sigma notation:

We can now write the series using sigma notation by summing the general formula for the odd numbers from 1 to 11: \(\sum_{n=1}^{11} (2n - 1)\) Thus, the given arithmetic series can be expressed in sigma notation as: \(\sum_{n=1}^{11} (2n - 1)\)

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Key Concepts

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

Arithmetic Series
An arithmetic series is a sequence of numbers in which each term after the first is obtained by adding a constant, referred to as the common difference, to the previous term. This common difference remains the same throughout the series. For instance, in the series 2, 4, 6, 8, the common difference is 2.

To notate an arithmetic series, we often use sigma notation, which provides a compact way to represent the sum of all terms within the series. The series expressed using sigma notation includes the general term of the sequence and indicates the starting and ending indices of the summation. In the context of the provided exercise, we are dealing with an arithmetic series of odd numbers with a common difference of 2, starting with 1 and ending with 21.
Sequence of Numbers
A sequence of numbers is an ordered list of numbers that follow a specific pattern or rule. The numbers in the sequence are known as terms. Sequences can be finite or infinite, depending on whether they have a last term or not. In many textbook problems, like the given exercise, we look for formulas to describe the n-th term of a sequence, which allows us to find any term's value without listing the entire sequence.

The formulation of a sequence is foundational in understanding its behavior and properties. For an arithmetic sequence, this formula incorporates the common difference, and for the series presented, an expression was derived that generates the nth odd number. Such explicit formulas enable us to transition from the sequence to the summation process using sigma notation.
Odd Number Sequence
An odd number sequence is a list of numbers where each term is an odd integer. Unlike even numbers, odd numbers cannot be divided evenly by 2. The sequence of odd numbers such as 1, 3, 5, 7, and so on, follows a specific arithmetic pattern where the common difference is 2. This is because adding 2 to an odd number always yields the next odd number.

In analyzing such a sequence, we observe that the first term is always 1, and every subsequent term can be found using the formula for the n-th term of an arithmetic sequence. Consequently, the n-th term of an odd number sequence can be represented as \(2n - 1\), with n starting from 1. It is essential to grasp this concept as it facilitates the understanding of more complex topics in number theory and series summation.

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Most popular questions from this chapter

Expand \(f(x)\) in powers of \(x\) $$f(x)=x \ln \left(1+x^{3}\right)$$

Find the Lagrange form of the remainder \(R_{n}(x)\). $$f(x)=\ln (1+x) ; \quad n=5$$

Suppose that the series \(\sum_{k=a}^{\infty} a_{i}(x-1)^{k}\)converges at \(x=3\) What can you conclude about the convergence or divergence of the following series? (a) \(\sum_{k=0}^{\infty} a_{k}\) (b) \(\sum_{n=0}^{\infty}(-1)^{k} a_{k}\) (c) \(\sum_{k=0}^{\infty}(-1)^{k} a_{k} 2^{k}\)

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Let \(\sum_{i=0}^{\infty} a_{z}\) be a convergent series and let \(R_{o}=\sum_{j=n+1}^{\infty} a_{k} .\) Prove that \(R_{n} \rightarrow 0\) as \(n \rightarrow \infty\). Note that if \(s_{n}\) is the nth partial sum of the series, then \(\sum_{k=9}^{\infty} a_{k}=s_{n}+R_{m} ; R_{n}\) is called the remainder.
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