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Chapter 9: Problem 5

Is a sequence or a series given? $$1-2+3-4+5-\cdots$$

Short Answer

Expert verified
The problem describes a series due to the presence of addition and subtraction between terms.

Step by step solution

01

Identify the Given Information

The problem presents a list of terms: \(1 - 2 + 3 - 4 + 5 - \cdots\). Each term in the list is interconnected through a "+" or "-" sign. To determine if this is a sequence or series, recognize how the terms are organized.
02

Define Sequence and Series

A sequence is a list of numbers arranged in an order following specific rules. A series is the sum of the terms of a sequence. In a sequence, terms are listed with commas (e.g., 1, 2, 3,...), while a series shows terms being added or subtracted (e.g., \(1 + 2 + 3 + ...\)).
03

Analyze the Problem Structure

Examine the problem closely to see if it lists numbers or expresses them as a sum or difference. The list \(1 - 2 + 3 - 4 + 5 - \cdots\) involves alternating addition and subtraction, suggesting the terms are added (or subtracted) together, forming a series.

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Key Concepts

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

Sequence
A sequence in mathematics is like a checklist of numbers arranged in a definite order. Each number, known as a term, follows a specific pattern or rule. Think of it as a number line where each step is dictated by a mathematical rule. For example:
  • Consider the sequence 2, 4, 6, 8,... where each number adds 2 to the previous one.
  • Another classic example is the Fibonacci sequence: 0, 1, 1, 2, 3, 5,... where each term is the sum of the two preceding ones.
Unlike in a series, the terms in a sequence are not summed up; they are simply listed. The visualization or understanding of a sequence helps in recognizing patterns and establishing the groundwork for discovering more complex mathematical phenomena.
Alternating Series
An alternating series is a special type of mathematical series where the signs of the terms alternate between positive and negative. This creates a zigzag pattern in the sum of the series. The sequence given in the original exercise is: \(1 - 2 + 3 - 4 + 5 - \cdots\). Here, every odd-numbered term is positive and every even-numbered term is negative.

Some properties of an alternating series:
  • The terms switch back and forth between addition and subtraction.
  • These series can sometimes converge (approach a specific value) if the terms get smaller with each step, according to the Alternating Series Test.
Because of its alternating pattern, it requires careful attention when analyzing its convergence or sum, and is a great example of how series can create order within apparent chaos.
Mathematical Series vs Sequence
Understanding the difference between a series and a sequence is key in calculus and advanced mathematics. This distinction helps in recognizing the type of mathematical treatment needed:

  • Sequence: A sequence is just a set of numbers ordered according to a rule. No arithmetic operations are implied among its elements.
  • Series: A series, on the other hand, takes the terms of a sequence and sums them up. It often involves complex analysis to determine if the series converges to a finite number or diverges.
In summary, every series stems from a sequence, but not every sequence can form a series with meaningful sums. For example, while a sequence might just list numbers like 1, 2, 3, 4,..., a series would mix these numbers up with '+' or '-' to make expressions like 1 + 2 + 3 + 4 +.... It's crucial for students to grasp this relationship to excel in both basic and advanced mathematics.

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Most popular questions from this chapter

(a) For a series \(\sum a_{n},\) show that \(0 \leq a_{n}+\left|a_{n}\right| \leq 2\left|a_{n}\right|\) (b) Use part (a) to show that if \(\sum\left|a_{n}\right|\) converges, then \(\sum a_{n}\) converges.

Determine whether the series converges. $$\sum_{n=1}^{\infty} \frac{(2 n) !}{(n !)^{2}}$$

Are true or false. Give an explanation for your answer. \(-5

Suppose that \(b_{n}>0\) for all \(n\) and \(\sum b_{n}\) converges. Show that if \(\lim a_{n} / b_{n}=0\) then \(\sum a_{n}\) converges.

Suppose \(0 \leq b_{n} \leq 2^{n} \leq a_{n}\) and \(0 \leq c_{n} \leq 2^{-n} \leq d_{n}\) for all \(n .\) Which of the series \(\sum a_{n}, \sum b_{n}, \sum c_{n},\) and \(\sum d_{n}\) definitely converge and which definitely diverge?
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