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Chapter 10: Problem 43

Determining Convergence or Divergence In Exercises \(17-44,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{(n !)^{2}}{(2 n) !}$$

Short Answer

Expert verified
The series converges by the Ratio Test, as the limit is \( \frac{1}{4} \lt 1 \).

Step by step solution

01

Understand the Series

The series to evaluate is given by \( \sum_{n=1}^{\infty} \frac{(n!)^2}{(2n)!} \). Our task is to determine whether this series converges or diverges.
02

Apply the Ratio Test

Apply the ratio test to determine convergence or divergence. For a series \( \sum a_n \), consider \( a_n = \frac{(n!)^2}{(2n)!} \). The ratio to evaluate is \( \lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| \).
03

Compute Ratio of Consecutive Terms

Determine \( \frac{a_{n+1}}{a_n} \) using:\[ a_{n+1} = \frac{((n+1)!)^2}{(2(n+1))!} = \frac{(n+1)^2(n!)^2}{(2n+2)(2n+1)(2n)!} \]Thus,\[ \frac{a_{n+1}}{a_n} = \frac{(n+1)^2 \cdot (2n)!}{(2n+2)(2n+1)(n!)^2} \]
04

Simplify the Ratio

Further simplify:\[ \frac{a_{n+1}}{a_n} = \frac{(n+1)^2}{(2n+2)(2n+1)} \]\(\frac{(n+1)^2}{(2n+2)(2n+1)}\) simplifies to \( \frac{(n+1)^2}{4n^2 + 6n + 2} \).
05

Evaluate the Limit

Compute the limit as \( n \to \infty \):\[ \lim_{n\to\infty} \frac{(n+1)^2}{4n^2 + 6n + 2} = \lim_{n\to\infty} \frac{n^2 + 2n + 1}{4n^2 + 6n + 2} \]Divide numerator and denominator by \( n^2 \):\[ \lim_{n\to\infty} \frac{1 + \frac{2}{n} + \frac{1}{n^2}}{4 + \frac{6}{n} + \frac{2}{n^2}} = \frac{1}{4} \]
06

Conclude with the Ratio Test

Since the limit \( \frac{1}{4} \lt 1 \), by the Ratio Test, the series \( \sum_{n=1}^{\infty} \frac{(n!)^2}{(2n)!} \) converges.

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Key Concepts

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

Ratio Test
The ratio test is a useful tool when trying to determine if an infinite series converges or diverges. It is particularly handy when dealing with series that include factorials, like the series in our exercise. Here's how it works:
  • Given a series \( \sum a_n \), observe the terms \( a_n \). In our exercise, \( a_n = \frac{(n!)^2}{(2n)!} \).
  • Calculate the ratio \( \left|\frac{a_{n+1}}{a_n}\right| \), which is basically the next term divided by the current term.
  • Take the limit of this ratio as \( n \) approaches infinity: \( \lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| \).
  • Evaluate this limit:
    • If the limit is less than 1, the series converges.
    • If the limit is greater than 1, the series diverges.
    • If the limit equals 1, the test is inconclusive.

In our example, solving this ratio eventually gave us \( \frac{1}{4} \), which is less than 1, indicating convergence.
Factorials in Series
Factorials, denoted by \( n! \), are products of all positive integers up to \( n \). They grow very quickly as \( n \) increases, often leading to large numbers in mathematical expressions. This rapid growth makes factorials both challenging and intriguing in analytical problems.

When dealing with series that contain factorials, like \( \sum \frac{(n!)^2}{(2n)!} \), understanding how they affect series is key:
  • Factorials compare relative sizes of numerators and denominators, influencing convergence or divergence.
  • In our exercise, the factorial in the denominator, \((2n)!\), grows significantly faster compared to the squared numerator \((n!)^2\).
  • This discrepancy in growth ensures that terms of the series get smaller as \( n \) increases, leading towards convergence.

Factorials provide a structured way of tackling combinatorics, probability, and series problems, offering insights into the behavior of sequences and series.
Infinite Series
An infinite series is an expression that sums an infinite number of terms. These appear frequently in calculus and are useful in modeling and approximating complex mathematical functions. An infinite series can be written in the form \( \sum_{n=1}^{\infty} a_n \).

Understanding convergence or divergence of such series involves:
  • Recognizing that convergent series approach a specific finite value as \( n \) increases.
  • If a series diverges, it either does not approach any particular value or grows infinitely.
  • Applying tests like the ratio test allows us to determine the behavior of infinite series efficiently.

In our exercise, we learned that applying the ratio test on the infinite series \( \sum_{n=1}^{\infty} \frac{(n!)^2}{(2n)!} \) shows convergence, offering a deeper understanding of how such series behave over infinite sums. Infinite series are vital tools in mathematical analysis and have applications across various scientific fields.

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Most popular questions from this chapter

How close is the approximation \(\sin x=x\) when \(|x|<10^{-3} ?\) For which of these values of \(x\) is \(x<\sin x ?\)

The Taylor series generated by \(f(x)=\sum_{n=0}^{\infty} a_{n} x^{n}\) is \(\sum_{n=0}^{\infty} a_{n} x^{n}\) A function defined by a power series \(\Sigma_{n=0}^{\infty} a_{n} x^{n}\) with a radius of convergence \(R>0\) has a Taylor series that converges to the function at every point of \((-R, R) .\) Show this by showing that the Taylor series generated by \(f(x)=\sum_{n=0}^{\infty} a_{n} x^{n}\) is the series \(\sum_{n=0}^{\infty} a_{n} x^{n}\) itself. \begin{equation} \begin{array}{c}{\text { An immediate consequence of this is that series like }} \\ {x \sin x=x^{2}-\frac{x^{4}}{3 !}+\frac{x^{6}}{5 !}-\frac{x^{8}}{7 !}+\cdots} \\ {\text {and}}\end{array} \end{equation} \begin{equation} x^{2} e^{x}=x^{2}+x^{3}+\frac{x^{4}}{2 !}+\frac{x^{5}}{3 !}+\cdots \end{equation} obtained by multiplying Taylor series by powers of \(x,\) as well as series obtained by integration and differentiation of convergent power series, are themselves the Taylor series generated by the functions they represent.

Computer Explorations Taylor's formula with \(n=1\) and \(a=0\) gives the linearization of a function at \(x=0 .\) With \(n=2\) and \(n=3\) we obtain the standard quadratic and cubic approximations. In these exercises we explore the errors associated with these approximations. We seek answers to two questions: \begin{equation} \begin{array}{l}{\text { a. For what values of } x \text { can the function be replaced by each }} \\ {\text { approximation with an error less than } 10^{-2} \text { ? }} \\ {\text { b. What is the maximum error we could expect if we replace the }} \\ {\text { function by each approximation over the specified interval? }}\end{array} \end{equation} Using a CAS, perform the following steps to aid in answering questions (a) and (b) for the functions and intervals in Exercises \(53-58 .\) \begin{equation} \begin{array}{l}{\text { Step } 1 : \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 } f^{(n+1)}(c) \text { associated }} \\ {\text { with the remainder term for each Taylor polynomial. }} \\ {\text { Plot 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 approximations }} \\ {\text { together. Discuss the graphs in relation to the information }} \\ {\text { discovered in Steps } 4 \text { and } 5 .}\end{array} \end{equation} $$f(x)=(1+x)^{3 / 2},-\frac{1}{2} \leq x \leq 2$$

In the alternating harmonic series, suppose the goal is to arrange the terms to get a new series that converges to \(-1 / 2 .\) Start the new arrangement with the first negative term, which is \(-1 / 2 .\) Whenever you have a sum that is less than or equal to \(-1 / 2,\) start introducing positive terms, taken in order, until the new total is greater than \(-1 / 2 .\) Then add negative terms until the total is less than or equal to \(-1 / 2\) again. Continue this process until your partial sums have been above the target at least three times and finish at or below it. If \(s_{n}\) is the sum of the first \(n\) terms of your new series, plot the points \(\left(n, s_{n}\right)\) to illustrate how the sums are behaving.

If \(\cos x\) is replaced by \(1-\left(x^{2} / 2\right)\) and \(|x|<0.5,\) what estimate can be made of the error? Does \(1-\left(x^{2} / 2\right)\) tend to be too large, or too small? Give reasons for your answer.
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