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Chapter 9: Problem 16

Verify that the infinite series diverges. \(\sum_{n=1}^{\infty} \frac{n !}{2^{n}}\)

Short Answer

Expert verified
Based on the ratio test, the given infinite series \(\sum_{n=1}^{\infty} \frac{n !}{2^{n}}\) diverges.

Step by step solution

01

Identify the terms and use the ratio test

The nth term of the series is \(\frac{n !}{2^{n}}\). The (n+1)th term is \(\frac{(n+1) !}{2^{n+1}}\). The limit of the ratio of the (n+1)th term to the nth term as n approaches infinity is \(\frac{(n+1) !/2^{n+1}}{n !/2^{n}}\). Simplifying, we get \(\frac{n+1}{2}\).
02

Calculate the limit

Taking the limit as n approaches infinity, we get \(\lim_{n \to \infty} \frac{n+1}{2}\). As n goes to infinity, \(\frac{n+1}{2}\) also goes to infinity.
03

Decision based on ratio test

Since the limit is greater than 1 (in fact, it is infinity), the given series \(\sum_{n=1}^{\infty} \frac{n !}{2^{n}}\) diverges as per the ratio test.

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Key Concepts

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

Infinite Series
An infinite series is the sum of an endless sequence of terms. Unlike a finite series, it does not stop at a certain number; it continues indefinitely. When looking at an infinite series, such as \[\sum_{n=1}^{\infty} \frac{n !}{2^{n}}\]we want to determine if it adds up to a finite number (converges) or grows indefinitely (diverges).
It's important because convergence or divergence can tell us about the behavior of certain functions and systems. Examples of infinite series include the geometric series, where each term is a constant ratio from the previous term, and the harmonic series, which is the sum of reciprocals of the natural numbers.
However, not all infinite series behave similarly, and different tests, like the Ratio Test, help determine their nature.
Ratio Test
The Ratio Test is a popular method for determining the convergence or divergence of an infinite series.
It examines the ratio of successive terms in a series. For a series whose nth term is denoted by \(a_n\), the Ratio Test involves calculating the limit:\[L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\]Here's how to interpret the result:
  • If \(L < 1\), the series converges.
  • If \(L > 1\) or \(L = \infty\), the series diverges.
  • If \(L = 1\), the test is inconclusive.
Applying the Ratio Test to the series \(\sum_{n=1}^{\infty} \frac{n !}{2^{n}}\), involve comparing the (n+1)th term:
\[\frac{(n+1) !}{2^{n+1}}\] to the nth term:
\[\frac{n !}{2^{n}}\] The ratio simplifies to \(\frac{n+1}{2}\). As \(n\) approaches infinity, this ratio gets larger and larger, leading to \(L = \infty\). Thus, by the Ratio Test, this series diverges.
Factorial
The factorial of a number is one of the key concepts in mathematics, denoted by \(n!\). It is the product of all whole numbers from \(n\) down to 1. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
Factorials grow rapidly. In our series, each term includes a factorial in the numerator, which significantly impacts the series' behavior.
This rapid growth is why factorials often appear in tests for convergence, such as in the Ratio Test.
  • Factorials are used in permutations and combinations, important in probability and statistics.
  • They appear in Taylor series, which approximate functions.
Understanding how factorials operate is crucial for solving and understanding problems dealing with series, especially when determining whether an infinite series like \(\sum_{n=1}^{\infty} \frac{n !}{2^{n}}\) diverges or converges.

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Most popular questions from this chapter

Prove that if the power series \(\sum_{n=0}^{\infty} c_{n} x^{n}\) has a radius of convergence of \(R\), then \(\sum_{n=0}^{\infty} c_{n} x^{2 n}\) has a radius of convergence of \(\sqrt{R}\).

Consider the sequence \(\left\\{a_{n}\right\\}=\left\\{\frac{1}{n} \sum_{k=1}^{n} \frac{1}{1+(k / n)}\right\\}\). (a) Write the first five terms of \(\left\\{a_{n}\right\\}\). (b) Show that \(\lim _{n \rightarrow \infty} a_{n}=\ln 2\) by interpreting \(a_{n}\) as a Riemann sum of a definite integral.

Fibonacci Sequence In a study of the progeny of rabbits, Fibonacci (ca. \(1170-\) ca. 1240 ) encountered the sequence now bearing his name. It is defined recursively by \(a_{n+2}=a_{n}+a_{n+1}, \quad\) where \(\quad a_{1}=1\) and \(a_{2}=1\). (a) Write the first 12 terms of the sequence. (b) Write the first 10 terms of the sequence defined by $$ b_{n}=\frac{a_{n+1}}{a_{n}}, \quad n \geq 1 . $$ (c) Using the definition in part (b), show that $$ b_{n}=1+\frac{1}{b_{n-1}} $$ (d) The golden ratio \(\rho\) can be defined by \(\lim _{n \rightarrow \infty} b_{n}=\rho\). Show that \(\rho=1+1 / \rho\) and solve this equation for \(\rho\).

Consider the sequence \(\left\\{a_{n}\right\\}\) where \(a_{1}=\sqrt{k}, a_{n+1}=\sqrt{k+a_{n}}\), and \(k>0\). (a) Show that \(\left\\{a_{n}\right\\}\) is increasing and bounded. (b) Prove that \(\lim _{n \rightarrow \infty} a_{n}\) exists. (c) Find \(\lim _{n \rightarrow \infty} a_{n^{\prime}}\)

Find all values of \(x\) for which the series converges. For these values of \(x\), write the sum of the series as a function of \(x\). $$ \sum_{n=1}^{\infty}(3 x)^{n} $$
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