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Chapter 8: Problem 37

Use the Ratio Test to determine the convergence or divergence of the given series. \(\sum_{n=1}^{\infty} \frac{(2 n) !}{\left(n ! \cdot 2^{n}\right)}\)

Short Answer

Expert verified
The series diverges.

Step by step solution

01

Identify the Sequence Terms

First, identify the terms of the sequence, which here are given by \( a_n = \frac{(2n)!}{(n! \cdot 2^n)} \). These are the terms whose convergence we want to test using the Ratio Test.
02

Apply the Ratio Test Formula

The Ratio Test involves calculating the ratio \( \lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right| \). Substitute the given sequence terms: \( a_{n+1} = \frac{(2(n+1))!}{((n+1)! \cdot 2^{n+1})} \) and simplify it to \( \frac{(2n+2)!}{((n+1)! \cdot 2^{n+1})} \).
03

Simplify the Ratio

Calculate \( \left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{(2n+2)! \cdot n! \cdot 2^n}{(2n)! \cdot (n+1)! \cdot 2^{n+1}} \right| \). This simplifies further to \( \frac{(2n+2)(2n+1)}{2(n+1)} \).
04

Evaluate the Limit

Now, find \( \lim_{{n \to \infty}} \frac{(2n+2)(2n+1)}{2(n+1)} \). Upon simplifying, we see \( \lim_{{n \to \infty}} \frac{4n^2 + 6n + 2}{2n + 2} = \lim_{{n \to \infty}} \frac{4n^2}{2n} \to 2n \). Thus, the limit is infinity.
05

Conclude Using the Ratio Test

As the limit from Step 4 is greater than 1, by the Ratio Test, the given series \( \sum_{n=1}^{\infty} \frac{(2n)!}{(n! \cdot 2^n)} \) diverges.

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Key Concepts

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

Convergence
Convergence in the context of series refers to the behavior of an infinite series where the sum approaches a finite number as the number of terms increases. When we discuss convergence, we are essentially checking if the terms of the series become increasingly smaller as we go further out in the series, thereby resulting in the total sum adding up to a specific value.
A series is said to be convergent if its terms approach zero as they continue to add up. This behavior means that although the series continues infinitely, the overall addition remains finite. Convergence plays a critical role in mathematical analysis and helps in understanding whether series can be summed to a specific number.
This property is pivotal when applying tests like the Ratio Test, which helps in checking whether a series meets these convergence criteria. Applying the Ratio Test involves examining the limit of the ratio between consecutive terms in the series to determine if they trend towards zero.
Divergence
Divergence occurs when the sum of an infinite series does not approach any particular limit, thus not settling down to any specific number. Instead, the series could potentially increase indefinitely, decrease indefinitely, or oscillate without reaching a stable sum. This behavior stands in stark contrast to convergence.
If a series diverges, its terms do not trend towards zero, and the accumulation of these terms does not stabilize at a finite point. This can mean that the partial sums of the series continue to grow larger as more terms are added.
Understanding divergence is crucial when evaluating series. The Ratio Test we applied to the series \(\sum_{n=1}^{\infty} \frac{(2n)!}{(n! \cdot 2^n)}\) showed divergence because the calculated limit was greater than one, indicating that the terms do not diminish adequately to allow for a finite summation.
Infinite Series
An infinite series is a sum of infinitely many terms. Notationally, it鈥檚 expressed with the sigma notation \(\sum_{n=1}^{\infty} a_n\), where \(a_n\) denotes the individual terms of the series. The concept of infinite series stretches across various domains of mathematics and is a fundamental component in calculus and analysis.
There are several pathways to consider when understanding series; one is recognizing whether an infinite series converges or diverges. This can be assessed via different tests and criteria, such as the Ratio Test, which investigates if the absolute value of the ratio of consecutive terms converges to less than one.
  • Convergent Infinite Series: These are series whose sum approaches a finite number.
  • Divergent Infinite Series: These are series whose sum does not stabilize.
Being able to identify the nature of an infinite series is essential for accurate mathematical grounding and application.
Factorials
Factorials are a mathematical operation denoted by an exclamation mark (!) used after a number. Essentially, \(n!\) represents the product of all positive integers from 1 to n. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). Factorials naturally arise in permutation calculations and other contexts requiring a comprehensive multiplication of descending integers.
In our series context, factorials play a significant role in the terms of the sequence. They contribute to defining the structure and behavior of the sequence's elements, such as \(\frac{(2n)!}{(n! \cdot 2^n)}\).
Factorials are notably crucial when calculating terms involving combinations and permutations, and they frequently appear in problems that have factorial ratios. Understanding how to manipulate factorials simplifies calculation and helps interpret the behavior of complex expressions or series.

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Most popular questions from this chapter

Use the Uniqueness Theorem to determine the coefficients \(\left\\{a_{n}\right\\}\) of the solution \(y(x)=\sum_{n=0}^{\infty} a_{n} x^{n}\) of the given initial value problem. \(d y / d x=1+x+y \quad y(0)=0\)

Consider the initial value problem $$ \frac{d y}{d x}=2-x-y, \quad y(0)=1 $$ \(\begin{array}{llll}\text { a. Calculate the power series } & \text { expansion }\end{array}\) \(y(x)=\sum_{n=0}^{\infty} a_{n} x^{n}\) of the solution up to the \(x^{7}\) term. b. Using the coefficients you have calculated, plot \(S_{3}(x)=\sum_{n=0}^{3} a_{n} x^{n}\) in the viewing rectangle \([-2,2] \times\) [-10,1.7] c. The exact solution to the initial value problem is \(y(x)=3-x-2 e^{-x},\) as can be determined using the methods of Section 7.7 (in Chapter 7 ). Add the plot of the exact solution to the viewing window. From the two plots, we see that the approximation is fairly accurate for \(-1 \leq x \leq 1\), but the accuracy decreases outside this subinterval. d. To see the improvement in accuracy that results from using more terms in a partial sum, replace the graph of \(S_{3}(x)\) with that of \(S_{7}(x)\)

In each of Exercises 49-54, use Taylor series to calculate the given limit. $$ \lim _{x \rightarrow 0} \frac{1-e^{2 x}}{x} $$

Determine whether the series converges absolutely, converges conditionally, or diverges. The tests that have been developed in Section 5 are not the most appropriate for some of these series. You may use any test that has been discussed in this chapter. \(\sum_{n=1}^{\infty}(-1)^{n} \frac{\ln (n)}{\sqrt{n}}\)

In each of Exercises 91-94 a function \(f\), a base point \(c\), and a point \(x_{0}\) are given. Plot \(y=\left|f^{(3)}(t)\right|\) for \(t\) between \(c\) and \(x_{0}\). Use your plot to estimate the quantity \(M\) of Theorem 2 . Then use your value of \(M\) to obtain an upper bound for the absolute error \(\left|R_{2}\left(x_{0}\right)\right|=\left|f\left(x_{0}\right)-T_{2}\left(x_{0}\right)\right|\) that results when \(f\left(x_{0}\right)\) is approximated by the order 2 Taylor polynomial with base point \(c\). $$ f(x)=e^{\sin (x)} \quad c=0 \quad x_{0}=0.4 $$
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