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Chapter 10: Problem 73

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{(2 n+2) !}{(2 n-1) !} $$

Short Answer

Expert verified
The sequence diverges to infinity.

Step by step solution

01

Understand the sequence

The given sequence is \(a_n = \frac{(2n + 2)!}{(2n - 1)!}\). We need to determine whether this sequence converges or diverges, and if it converges, find the limit. A factorial, \((k)!,\) is the product of all positive integers up to \(k\).
02

Simplify the sequence

Expand the factorials in the sequence: \( (2n + 2)! = (2n + 2)(2n + 1)(2n)(2n - 1)! \). The sequence \(a_n\) simplifies to:\[a_n = \frac{(2n + 2)(2n + 1)(2n)(2n - 1)!}{(2n - 1)!} = (2n + 2)(2n + 1)(2n).\]
03

Determine the behavior as \(n\) approaches infinity

Examine the simplified form of \(a_n = (2n + 2)(2n + 1)(2n)\). As \(n\) approaches infinity, these terms grow without bound. Therefore, \(a_n\) becomes infinitely large.
04

Evaluate convergence or divergence

Since \(a_n\) tends to infinity as \(n\) tends to infinity, it does not approach a finite number. Hence, the sequence does not converge, but instead diverges.

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Key Concepts

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

Factorial
Factorials are fundamental in understanding sequences and series in mathematics. If you see a symbol like \(n!\), this reads as "n factorial."
In simple terms, \(n!\) is the product of all positive integers from 1 up to n. For example, \(4! = 4 \times 3 \times 2 \times 1 = 24\). This concept is greatly useful when dealing with permutations, probabilities, and exponential growth as it helps describe various patterns.
  • The factorial grows very fast with larger values of n. For example, even \(10! = 3,628,800\), which is a massive number.
  • Factorials play a key role in simplifying terms within sequences like \(a_n = \frac{(2n+2)!}{(2n-1)!}\).
Understanding how to expand and simplify factorials is crucial when determining sequence behavior. It helps break down complex expressions into manageable forms, revealing deeper insights into their behavior as values of n increase.
Convergent Sequence
A convergent sequence is one in which the terms get closer and closer to a certain value as the sequence progresses. To say a sequence converges, means essentially that it approaches a "limit" value.
You can picture it as a sequence "settling down" to a particular value and staying there as n becomes very large.
  • This limit is finite and real.
  • An example of a convergent sequence is a constant sequence, where every term is the same.
  • Sequences determined to converge are predictable and stable.
Determining convergence often involves simplifying the sequence and checking if, as n grows infinitely, the sequence approaches a specific value. This is not the case with our \(a_n\) as given, because it doesn't settle at a specific number.
Divergent Sequence
In contrast to convergent sequences, divergent sequences never approach a single limit, and often grow without bound. In a more straightforward sense, terms within a divergent sequence don't "settle" on a value.
A divergent sequence can become infinitely large or display erratic behavior without getting closer to any specific number.
  • Our analyzed sequence \(a_n = (2n+2)(2n+1)(2n)\) is divergent as it grows endlessly when n approaches infinity.
  • Divergence can also occur if a sequence's terms oscillate, continually moving further away from a potential limit.
  • To declare a sequence as divergent, one demonstrates lack of a limiting value as n approaches infinity.
Identifying divergence helps in understanding the behavior of functions, especially in mathematical modeling or in complex calculus problems. Being able to pinpoint whether a sequence converges or diverges is crucial for deeper mathematical analysis.

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Most popular questions from this chapter

Which of the series in Exercises 13 46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) $$ \sum_{n=1}^{\infty} \operatorname{sech} n $$

Use a CAS to perform the following steps for the sequences. a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit \(L\) ? b. If the sequence converges, find an integer \(N\) such that \(\quad\left|a_{n}-L\right| \leq 0.01\) for \(n \geq N .\) How far in the sequence do you have to get for the terms to lie within 0.0001 of \(L ?\) $$ a_{n}=n \sin \frac{1}{n} $$

\(\text { a.}\) Suppose that \(f(x)\) is differentiable for all \(x\) in \([0,1]\) and that \(f(0)=0 .\) Define sequence \(\left\\{a_{n}\right\\}\) by the rule \(a_{n}=n f(1 / n)\) Show that \(\lim _{n \rightarrow \infty} a_{n}=f^{\prime}(0) .\) Use the result in part (a) to find the limits of the following sequences \(\left\\{a_{n}\right\\} .\) $$ \begin{array}{l}{\text { b. } a_{n}=n \tan ^{-1} \frac{1}{n} \quad \text { c. } a_{n}=n\left(e^{1 / n}-1\right)} \\ {\text { d. } a_{n}=n \ln \left(1+\frac{2}{n}\right)}\end{array} $$

Two complex numbers \(a+i b\) and \(c+i d\) are equal if and only if \(a=c\) and \(b=d .\) Use this fact to evaluate \begin{equation}\int e^{a x} \cos b x d x \text { and } \int e^{a x} \sin b x d x\end{equation} from \begin{equation}\int e^{(a+i b) x} d x=\frac{a-i b}{a^{2}+b^{2}} e^{(a+i b) x}+C\end{equation} where \(C=C_{1}+i C_{2}\) is a complex constant of integration.

\begin{equation} \begin{array}{l}{\text { Using a CAS, perform the following steps to aid in answering }} \\ {\text { questions (a) and (b) for the functions and intervals in Exercises }} \\ {57-62 .}\\\\{\text { Step } I : \text { Plot the function over the specified interval. }} \\ {\text { Step } 2 : \text { Find the Taylor polynomials } P_{1}(x), P_{2}(x), \text { and } P_{3}(x) \text { at }} \\ {x=0}\\\\{\text { Step } 3 : \text { Calculate the }(n+1) \text { st derivative } \int^{(n+1)}(c) \text { associat- }} \\ {\text { ed with the remainder term for each Taylor polynomial. Plot }} \\ {\text { the derivative as a function of } c \text { over the specified interval }} \\ {\text { and estimate its maximum absolute value, } M .}\\\\{\text { Step } 4 : \text { Calculate the remainder } R_{n}(x) \text { for each polynomial. }} \\ {\text { Using the estimate } M \text { from Step } 3 \text { in place of } f^{(n+1)}(c), \text { plot }} \\ {R_{n}(x) \text { over the specified interval. Then estimate the values of }} \\ {x \text { that answer question (a). }}\\\\{\text { Step } 5 : \text { Compare your estimated error with the actual error }} \\ {E_{n}(x)=\left|f(x)-P_{n}(x)\right| \text { by plotting } E_{n}(x) \text { over the specified }} \\ {\text { interval. This will help answer question (b). }} \\ {\text { Step } 6 : \text { Graph the function and its three Taylor approxima- }} \\ {\text { tions together. Discuss the graphs in relation to the informa- }} \\ {\text { tion discovered in Steps } 4 \text { and } 5 .}\end{array} \end{equation} $$f(x)=(\cos x)(\sin 2 x), \quad|x| \leq 2$$
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